V1.0 Introduction to the course
A brief introduction to this course, Introductory Quantum Mechanics. The set of videos is aimed at second-year physics undergraduates.
Notes and problems to accompany the videos are available at:
felixflicker.com/teaching
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Hello and welcome to this introductory
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video for introductory quantum mechanics
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the set of videos is going to cover a
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second year course in quantum mechanics.
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I'll assume knowledge of a standard
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first year undergraduate physics course
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so subjects such as complex numbers
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differential equations
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vectors and matrices classical mechanics
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and so on
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I'll provide problem sets and notes to
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accompany the videos
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on my website the link to which can be
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found in the youtube channel description
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I'll try to mix things up a little bit
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sometimes I'll be walking along like
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this with my dog Geoffrey
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sometimes I will instead
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be writing at this board sometimes
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i will be recording worked examples
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in a little bit more detail using pen
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and paper
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let's switch back to the field
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with regard to youtube I don't know
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whether you're seeing adverts at the
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start of this video
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if you don't wish to see the adverts you
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can use something such as adblock plus
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which is available
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for all browsers which will stop the
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adverts on youtube
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there's a setting in the bottom right
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hand corner
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which will allow you to change the
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quality of the video the videos are all
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filmed in 1080p
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fully high definition so if at any point
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you can't see what's being written on
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the board for example
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try increasing the quality setting and
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hopefully that should make things
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clearer
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and finally also in the settings
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you'll find that you can increase the
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speed of the videos up to
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two times so if you want to watch things
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a bit faster that should be possible
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so thank you very much for your time and
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I look forward to seeing you in the
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coming videos
V1.1 History of quantum mechanics
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
a brief history of the experimental developments which necessitated the development of quantum mechanics.
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hello in this video i'm going to give a
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brief
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overview of the history of the subject
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of quantum mechanics
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so quantum mechanics is somewhat unique
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amongst physics courses
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in seemingly requiring a historical
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background
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and i should add as a disclaimer at the
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start that i'm not a historian i'm a
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physicist
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and so you may want to go and check
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these facts yourself
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nevertheless the purpose of the video
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and the take home message from it
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is that however weird things get later
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on in the course
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quantum mechanics is firmly rooted in
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experimental observation
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it wasn't that we thought physics needed
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to get more magical so we started
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inventing strange interpretations of
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things
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what happened is that we carried out a
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set of experiments which revealed to us
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that the world is indeed more magical
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and we invented quantum mechanics in
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order to explain those observations
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so to begin with some pre-history
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in 1756 flame tests were developed
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in which you take a pure element and
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heat is over a flame and the flame
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changes colour
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it's an experiment you may have done
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yourself
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the this is now understood to be a
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result of the discrete nature of the
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atomic levels
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electrons occupy within atoms so the
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word quantum in quantum mechanics refers
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to
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discrete or separate and it's what
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happens when we
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go from our large scale macroscopic
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world which is seemingly continuous
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and go down to the smallest scale we
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kind of knew things had to get
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discrete somehow. In 1801 Thomas Young
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carried out an experiment which is now
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called Young's slits
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or the two-slit experiment in which he
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took two
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thin closely spaced slits in a piece of
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card shone light through it
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and measured a pattern on a screen and saw
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that light is able to interfere
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he took this as good evidence of the
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wave nature of light
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we'll take a look at that interference
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pattern in another experiment
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in 1850 atomic line spectra were measured
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for the first time
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we take a gas for pure element and
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pass white light through it
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and find that a set of discrete
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frequencies will be removed from the
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white light when we take that same gas
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and heat it up we see the light that's
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emitted from it is exactly that same set
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of frequencies that were
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absorbed from the light that was passed
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through it we now know this to be
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evidence
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of the discrete nature of the atomic
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levels once again
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in 1887 the
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photoelectric effect was measured by
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Heinrich Hertz
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so some gold leaf is taken and charged
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electrically
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two bits of gold leaf
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would repel from one another
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when light is shone onto the gold leaf we
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see that the charge dissipates
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the thing that was difficult to explain
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classically was that
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the frequency of the light being shone
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onto the gold leaf
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has to be above a certain threshold
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frequency there was no classical
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explanation of this
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in 1897 J J Thompson and others
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measured the behaviour of cathode rays
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so initially you might have been
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forgiven for thinking cathode rays were
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something like
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beams of light they were rays which
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could light up
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gases such as argon but unlike light
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it was possible to bend these rays using
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magnetic
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fields so what this showed us is that
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those rays were being carried
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by massive particles hence their ability
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to be deflected by magnetic fields and
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accelerated
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and so this was the first evidence for
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subatomic particles
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in this case electrons.
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We've gone
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through a set of experiments we've taken
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this world that appears
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on our everyday scales to be continuous
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and we've observed that down on the
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smaller scales that continuity
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emerges out of small
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sets of discrete things
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in 1911 the Millikan experiment was
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carried out in which
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oil drops electrically charged could be
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suspended using
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electrostatic potentials by measuring
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the required potential to levitate an
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oil drop
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it's possible to measure the charge
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accurately and it was found
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by Millikan that the charge always came
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in
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an integer multiple of some smallest
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amount which we now understand to be the
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charge of the electron
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in 1913 the Rutherford experiment was
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carried out also known as the
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Geiger-Marsden experiment after the
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people who really did most of the work
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in which alpha particles were seen to
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deflect
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from atoms alpha particles we now
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understand to be the nuclei
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of helium four atoms so
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occasionally these alpha particles would
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deflect through more than 90 degrees
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reflecting back
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so this was evident that while the atom
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is overall charged neutral
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that charge is not evenly distributed
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there's a small positively charged
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nucleus and a negative charge around
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that core
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this led to a problem for classical
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physics because if the
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distribution is as described by the
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Rutherford experiment
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why don't the negative charges fall into
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the positive charges to minimize their
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energy
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there's then a short break in the
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experiments i'd like to mention owing
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to the world war and world pandemic
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and the next i want to mention is the
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Compton scattering experiment in 1923.
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This saw the scattering of light
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by electrons and is used as evidence
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that much like we've seen both the
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particle and wave-like nature to light
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you could see a particle and wave like
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nature to electrons
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in 1923 to 1927 the Davisson-Germer
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experiment
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saw interference patterns in the
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electrons deflecting off the surface of
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nickel
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so this was further evidence for the
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fact that particles such as the electron
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can have wave-like characteristics
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so that's a set of experiments that
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does not exhaust the set of experiments
0:05:48.639,0:05:52.000
that require quantum explanations rather
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than classical
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but it's some of the key examples
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coincident with this of course were
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theoretical developments.
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Quantum mechanics has a very nice
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history of theory and experiment working
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together
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so the pre-history i think one of the
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most important things to mention before
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the development of quantum mechanics
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was Maxwell's equations you the unified
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theory of electromagnetism
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and in particular the prediction of
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electromagnetic waves so
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Maxwell's equations were written down
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in around
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1865 something like that
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and while coming before special
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relativity and quantum mechanics
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they're completely compatible with both
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so
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what this means is that light emits both
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a classical description
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which is wholly accurate provided
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there are no interactions so
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non-interacting light can be described
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completely classically in terms of waves
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but it can also be described completely
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quantum mechanically in terms of
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particles which we now call photons
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so this is a useful trick that'll allow
0:06:48.960,0:06:53.280
us to do various experiments
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in our own home of a quantum mechanical
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nature using light
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in 1900 Lord Rayleigh
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and shortly after James Jeans in 1905
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with Rayleigh
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developed a theory of the spectral
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radiance of black bodies
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so what this means is the
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set of frequencies coming off a body
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at a given temperature
0:07:16.560,0:07:19.759
so all bodies emit black body
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radiation
0:07:19.759,0:07:23.360
it's just that good absorbers are good
0:07:21.599,0:07:25.680
emitters and so
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black things such as pure carbon for
0:07:25.680,0:07:28.639
example
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being very good absorbers give very
0:07:28.639,0:07:33.680
clear black body radiation spectra
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now the Rayleigh-Jeans law as it's
0:07:33.680,0:07:37.919
called predicted that the spectral
0:07:36.160,0:07:39.680
radiance would continue to increase with
0:07:37.919,0:07:41.039
increasing frequency
0:07:39.680,0:07:43.199
this led to what's called the
0:07:41.039,0:07:43.599
ultraviolet catastrophe a prediction
0:07:43.199,0:07:45.440
that
0:07:43.599,0:07:46.800
all bodies should effectively have an
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infinite amount of energy in them that
0:07:46.800,0:07:49.520
they're giving off
0:07:47.599,0:07:51.280
as electromagnetic radiation which of
0:07:49.520,0:07:52.479
course was not what was experimentally
0:07:51.280,0:07:54.960
observed
0:07:52.479,0:07:55.759
in 1900 Max Planck came up with what's
0:07:54.960,0:07:58.319
really the first
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truly quantum theory where he came up
0:07:58.319,0:08:03.280
with a phenomenological model
0:08:00.240,0:08:05.680
which gave a very good match to the
0:08:03.280,0:08:07.599
experimentally observed black body
0:08:05.680,0:08:10.800
radiation spectra
0:08:07.599,0:08:14.479
and he did so by saying that the
0:08:10.800,0:08:16.080
light had was being emitted only in
0:08:14.479,0:08:18.720
discrete packets that he called
0:08:16.080,0:08:21.199
quanta and he identified that
0:08:18.720,0:08:23.199
the energy of one of these packets for a
0:08:21.199,0:08:26.319
wave with angular frequency omega
0:08:23.199,0:08:27.680
was given by hbar omega where hbar is
0:08:26.319,0:08:29.759
now what we call the reduced Planck's
0:08:27.680,0:08:33.599
constant
0:08:29.759,0:08:33.599
so he identified that pre-factor
0:08:34.240,0:08:38.560
his model was phenomenological meaning
0:08:36.080,0:08:40.479
that it fit the data accurately
0:08:38.560,0:08:41.839
but there was no microscopic explanation
0:08:40.479,0:08:44.800
as to where that
0:08:41.839,0:08:46.080
expression came from Plank himself
0:08:44.800,0:08:47.120
believed it just to be a mathematical
0:08:46.080,0:08:49.920
trick
0:08:47.120,0:08:51.760
so in 1905 the major development was
0:08:49.920,0:08:54.000
made by Albert Einstein
0:08:51.760,0:08:55.440
in a year in which he wrote
0:08:54.000,0:08:58.320
four papers each of which
0:08:55.440,0:08:58.800
individually revolutionized physics so
0:08:58.320,0:09:01.680
he
0:08:58.800,0:09:03.279
outlined special relativity at the same
0:09:01.680,0:09:03.920
time he explained the photoelectric
0:09:03.279,0:09:06.399
effect
0:09:03.920,0:09:07.200
by taking Planck's hypothesis seriously
0:09:06.399,0:09:09.360
and saying
0:09:07.200,0:09:11.040
rather than a mathematical tool this is
0:09:09.360,0:09:13.120
a physical statement light is really
0:09:11.040,0:09:15.600
conveyed by individual packets
0:09:13.120,0:09:16.720
quanta which we now call photons and the
0:09:15.600,0:09:18.720
energy of one of these
0:09:16.720,0:09:21.839
photons is hbar times the angular
0:09:18.720,0:09:24.399
frequency of that photon
0:09:21.839,0:09:24.399
he also
0:09:25.120,0:09:28.240
provided convincing evidence of the
0:09:27.120,0:09:30.160
atomic theory
0:09:28.240,0:09:32.560
by explaining Brownian motion of pollen
0:09:30.160,0:09:34.640
molecules in water as
0:09:32.560,0:09:36.320
them jostling around from the impacts of
0:09:34.640,0:09:37.440
individual atoms and this led to the
0:09:36.320,0:09:40.480
widespread
0:09:37.440,0:09:40.959
adoption of the atomic theory and he
0:09:40.480,0:09:42.880
also
0:09:40.959,0:09:44.720
in a fourth paper wrote down probably
0:09:42.880,0:09:47.120
the most famous equation ever
0:09:44.720,0:09:49.279
E=mc^2
0:09:47.120,0:09:50.399
in 1911 the same year as the Millikan
0:09:49.279,0:09:53.760
experiment
0:09:50.399,0:09:55.279
Niels Bohr wrote down the quantum
0:09:53.760,0:09:56.959
theory of the atom
0:09:55.279,0:09:58.480
it was again a phenomenological theory
0:09:56.959,0:10:02.240
but he found an equation
0:09:58.480,0:10:04.399
which predicted accurately the
0:10:02.240,0:10:06.079
measurements made in the atomic line
0:10:04.399,0:10:07.600
spectra for example
0:10:06.079,0:10:09.519
in terms of the electrons occupying
0:10:07.600,0:10:11.279
discrete energy levels in the atom
0:10:09.519,0:10:12.640
well we now know the Bohr model to be
0:10:11.279,0:10:14.399
incorrect but
0:10:12.640,0:10:16.079
it made accurate predictions for the
0:10:14.399,0:10:17.839
energy levels and it was really the
0:10:16.079,0:10:19.680
motivation that led to the development
0:10:17.839,0:10:22.160
of quantum theory later on
0:10:19.680,0:10:23.440
so 1911 leads to the end of what we call
0:10:22.160,0:10:25.600
old quantum theory
0:10:23.440,0:10:27.360
a set of phenomenological ideas that is
0:10:25.600,0:10:30.079
things that make good predictions
0:10:27.360,0:10:31.760
in terms of experimental observations
0:10:30.079,0:10:33.040
but which don't have a microscopic
0:10:31.760,0:10:34.800
explanation as to where they're coming
0:10:33.040,0:10:36.399
from
0:10:34.800,0:10:39.760
the next major development was
0:10:36.399,0:10:43.839
by Louis de Broglie in 1923
0:10:39.760,0:10:45.839
who hypothesized that just as
0:10:43.839,0:10:47.920
traditionally wave-like things such as
0:10:45.839,0:10:49.120
light admit a particle description as
0:10:47.920,0:10:51.040
Einstein said
0:10:49.120,0:10:53.120
perhaps traditionally
0:10:51.040,0:10:55.279
particle-like things such as electrons
0:10:53.120,0:10:56.560
admit a wave-like description and he
0:10:55.279,0:10:58.000
proposed a formula
0:10:56.560,0:11:00.079
telling us that the momentum of the
0:10:58.000,0:11:00.959
particle is linearly related to the wave
0:11:00.079,0:11:02.720
vector
0:11:00.959,0:11:04.079
of the corresponding wave and the
0:11:02.720,0:11:06.079
constant of proportionality is the
0:11:04.079,0:11:08.399
reduced Planck's constant
0:11:06.079,0:11:10.320
this brings us to 1925 which is pretty
0:11:08.399,0:11:12.959
much where this course will get us to
0:11:10.320,0:11:14.240
so in 1925 Erwin Schroedinger wrote down
0:11:12.959,0:11:17.360
the Schroedinger equation
0:11:14.240,0:11:19.200
as part of what is called wave mechanics
0:11:17.360,0:11:20.720
in the same year Werner Heisenberg and
0:11:19.200,0:11:23.680
others including Niels Bohr
0:11:20.720,0:11:24.320
wrote down matrix mechanics and these
0:11:23.680,0:11:26.959
were two
0:11:24.320,0:11:28.320
microscopic models for how particles
0:11:26.959,0:11:30.800
behave
0:11:28.320,0:11:32.480
down on the smallest scales so this has
0:11:30.800,0:11:34.320
gone beyond a phenomenological model to
0:11:32.480,0:11:36.399
give a microscopic explanation as to why
0:11:34.320,0:11:38.880
things are happening
0:11:36.399,0:11:39.920
later on that year Schroedinger
0:11:38.880,0:11:40.720
showed the equivalence of the two
0:11:39.920,0:11:42.640
approaches
0:11:40.720,0:11:44.640
uniting wave mechanics and matrix
0:11:42.640,0:11:45.920
mechanics into what we now call quantum
0:11:44.640,0:11:48.480
mechanics
0:11:45.920,0:11:50.000
so that's as far as this course is going
0:11:48.480,0:11:52.240
to take us there are of course
0:11:50.000,0:11:54.800
many further development i think the
0:11:52.240,0:11:56.079
major one being in 1935 when Einstein
0:11:54.800,0:11:57.600
Podolsky and Rosen
0:11:56.079,0:11:59.279
developed what's now called the EPR
0:11:57.600,0:12:01.440
paradox which led
0:11:59.279,0:12:03.120
after John Stuart Bell came up with a
0:12:01.440,0:12:05.760
testable prediction for it
0:12:03.120,0:12:07.440
and in 1980 Alain Aspect carried out a
0:12:05.760,0:12:09.600
set of experiments confirming
0:12:07.440,0:12:11.680
that quantum entanglement is a
0:12:09.600,0:12:13.440
fundamental property of the universe
0:12:11.680,0:12:15.600
arguably there are only two truly
0:12:13.440,0:12:17.360
fundamentally quantum effects
0:12:15.600,0:12:18.639
one is wave particle duality which we'll
0:12:17.360,0:12:20.079
see a lot of in this course
0:12:18.639,0:12:22.480
and the other is quantum entanglement
0:12:20.079,0:12:24.720
which we'll see less of.
0:12:22.480,0:12:26.399
While we'll see many other things such as
0:12:24.720,0:12:28.560
for example the Heisenberg uncertainty
0:12:26.399,0:12:30.320
principle which are undoubtedly quantum
0:12:28.560,0:12:32.639
in all the other cases other than wave
0:12:30.320,0:12:34.639
particle duality and entanglement
0:12:32.639,0:12:35.680
there are classical precedents that we
0:12:34.639,0:12:37.360
can find
0:12:35.680,0:12:38.800
it's not that quantum mechanics is less
0:12:37.360,0:12:40.480
magical the more that you study it
0:12:38.800,0:12:42.800
it's just that classical mechanics
0:12:40.480,0:12:48.639
was more magical than we gave it
0:12:42.800,0:12:48.639
credit for okay thank you for your time
V1.2 The Schroedinger equation
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
the time-dependent Schroedinger equation (TDSE), and how to obtain from it the time-independent Schroedinger equation (TISE).
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hello the behaviour of particles in
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quantum mechanics is governed by what's
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called the schrodinger equation
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written here in its time-dependent form
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so we sometimes call this
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the time-dependent schrodinger equation
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usually abbreviated to TDSE for short
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it was written down in 1925
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by this man Erwin Schrodginer as you can
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see a very well-dressed gentleman as
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they
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were back in the 1920s but you know
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what, it's the
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20s again so let's see if we can't
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conjure up a bit of that 1920s chic
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in this video
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it worked. Magic! OK.
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So the schrodinger equation when
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schrodinger originally wrote it down
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was intended to be a classical
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description of classical waves
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this quantity psi which we call the wave
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function was thought of
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as something like the wave function
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which would describe say water waves
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in which case it would take some value
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which would tell us about the height of
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the water wave
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as a function of position and time
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in fact as we'll see while there is a
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good description of
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the wave function as something like a
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classical wave, in fact quantum mechanics
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goes beyond classical physics as we now
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of course know. So before we
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proceed let me just make a couple of
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notational points
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when i write d(psi)/dt like this, this
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is a partial derivative with respect to
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time
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holding all spatial coordinates constant
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so we can also denote this in the
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following way
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so we denote this
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stating explicitly that
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the positions are all constant
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sometimes i'll denote this
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in a simpler manner in the following
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form
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where partial subscript t is just
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shorthand for
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d by dt acting on psi and sometimes i'll
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use
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newton's notation psi dot
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similarly a partial derivative with
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respect to x
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implies that we're holding the other
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variables constant so time
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y and z which can again be abbreviated
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just partial subscript x acting on
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psi
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and sometimes we'll write this as psi
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prime
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where this last notation will only
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really be used in one dimension where
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it's unambiguously a derivative with
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respect to x
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so there are a couple of key
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assumptions we're going to make
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throughout this course let's just wipe
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this board off and write them down
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when i say wipe the board off of course
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i mean magically clear the board
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so let's write down some key assumptions
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the first is that the schrodinger
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equation
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is non-relativistic. It's going to
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describe the behavior of some massive
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particle
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in the non-relativistic limit. There are
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relativistic extensions to it
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but we're not going to consider those in
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this course
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the second is that we're only going to
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describe single particles.
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Again there are extensions to the
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schrodinger equation which will allow it
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to deal with multiple particles
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but that's not going to be the focus of
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this course. The wave function psi
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always governs the behavior of a single
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particle
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okay so let's take a closer look at
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what's going on in the schrodinger
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equation let's clear the board again
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so let's have the schrodinger equation
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back
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so we'll look at properties of the wave
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function psi in future videos
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but for now let's just note that it's a
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complex-valued function
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and it has out the front of it an
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arbitrary
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global complex phase.
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It's a mathematical redundancy built
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into the the wave function
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it has this global phase which can be
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changed arbitrarily.
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But the relative phase between two
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different wave functions
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is important and we will see that later
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on in the course
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but for now there is this redundancy
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built into it.
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We've got
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the imaginary unit here i
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square root minus one. Imaginary
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numbers play an important role in
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quantum mechanics.
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hbar here is the reduced planck's
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constant h over 2 pi
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with units of energy multiplied by time
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and then we have this quantity here H
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which
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is called the hamiltonian let's write
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that down
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so h is defined as follows
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it's a differential
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operator we have this nabla squared in
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here
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which okay we can expand as follows
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so it's just the partial derivative
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with respect to x
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squared plus that of y squared and that
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of z squared remembering the notation
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from the last board and if this looks
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confusing just remember this is always
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acting on psi so this term here
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(d/dx)^2 acting on psi is really just
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psi
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double prime. Okay.
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So it's called the Hamiltonian
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and it's what's called an energy
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operator for the system
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so the hamiltonian acting on psi
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is going to return the energy of the
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system as we'll see
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in a second in general
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specifying the hamiltonian specifies the
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entire quantum problem we want to solve
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so when we
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write down a quantum mechanical
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description of a system we just need to
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write down the hamiltonian
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and in fact this first term is fixed
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what we what need to specify is the
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potential of the system which is much
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like what you do in classical mechanics
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as well
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so the potential we will assume in
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this course is time independent
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it doesn't need to be but we won't
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consider
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time dependent potentials here. So
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let's take a look at attempts to
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deal with
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how to solve this equation
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so the time-dependent schrodinger
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equation
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where i've written it with psi dot this
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time is a separable
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equation what we mean by this
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is that we can substitute the
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following anzatz
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we can say that psi(x,t)
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is equal to:
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-- and let's treat it in three
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dimensions in general -- is equal to
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phi(x) only
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and T(t) only
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so these are two separate functions one
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of which is only a function of time
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and one of which is only a function of
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position when we substitute
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that in
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we see that the equation reduces to the
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following form
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where the phi has pulled through the
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time derivative
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because it's not a function of time and
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this is a partial derivative
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the partial derivative of T(t) with respect
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to time
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is actually a total derivative because
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it's only a function of time
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and over here the hamiltonian because the
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potential is time independent
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hamiltonian only acts on the spatial
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coordinates so we can pull through the t
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over here rearranging slightly we get
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two different equations
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one purely in terms of t and one purely
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in terms of x
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so we've turned what was a partial
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differential equation in terms of both x
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and t
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into two separate equations one is an
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00:07:22,479 --> 00:07:24,400
ordinary differential equation in terms
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of time
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and the other is potentially still a
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partial differential equation in terms
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of positions
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so the fact that these two are equal for
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all times
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and all positions means they must both
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be equal to the same constant
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and we're going to suggestively call
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that constant E
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suggestive because it should remind us
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of an energy
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so let's number these equations let's
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call this one (i)
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and this one (ii)
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we'll move this up to the top of the
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board and we'll deal with (i) and (ii)
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00:07:54,479 --> 00:07:59,440
separately
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so (i) simply rearranges
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00:07:59,440 --> 00:08:02,319
to the following form
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00:08:02,800 --> 00:08:11,120
H acting on phi is equal to
233
00:08:08,400 --> 00:08:12,080
E multiplying phi and this is what's
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called the time
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00:08:12,080 --> 00:08:17,120
independent schrodinger equation because
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00:08:15,520 --> 00:08:19,120
phi here is now only a function of
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00:08:17,120 --> 00:08:22,639
positions not of time.
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00:08:19,120 --> 00:08:24,639
Let's put a box around it in general
239
00:08:22,639 --> 00:08:27,120
this one is very tricky to solve in fact
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in general it's impossible to solve
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00:08:27,120 --> 00:08:30,560
remember the hamiltonian here contains
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00:08:28,720 --> 00:08:32,159
all the real information in the problem
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in the form of the potential
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so in general we need to solve this
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00:08:34,080 --> 00:08:38,080
differential equation
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00:08:36,320 --> 00:08:39,440
in terms of boundary conditions which we
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00:08:38,080 --> 00:08:42,000
specify and
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00:08:39,440 --> 00:08:42,800
you can see that it's an eigenvalue
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00:08:42,000 --> 00:08:45,440
equation
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00:08:42,800 --> 00:08:45,839
in that we need to solve this for
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00:08:45,440 --> 00:08:48,080
both
252
00:08:45,839 --> 00:08:49,519
the eigenfunctions phi(x) and the
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00:08:48,080 --> 00:08:50,560
eigenenergies E
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00:08:49,519 --> 00:08:52,800
which will be the energies of the
255
00:08:50,560 --> 00:08:55,279
particle let's take a look at the second
256
00:08:52,800 --> 00:08:57,519
equation there
257
00:08:55,279 --> 00:08:58,880
where i've done our favorite trick of
258
00:08:57,519 --> 00:09:01,920
multiplying through by
259
00:08:58,880 --> 00:09:05,760
dt. We can then integrate both sides
260
00:09:01,920 --> 00:09:09,360
to give the result:
261
00:09:05,760 --> 00:09:11,200
that is, the time evolution of t is simply a
262
00:09:09,360 --> 00:09:12,160
phase winding. The winding of the complex
263
00:09:11,200 --> 00:09:14,240
phase
264
00:09:12,160 --> 00:09:16,480
so sticking our anzatzes back together
265
00:09:14,240 --> 00:09:19,839
again remember psi(x,t) the wave function
266
00:09:16,480 --> 00:09:22,240
is the product of T(t) and phi(x) we find that
267
00:09:19,839 --> 00:09:24,320
if we can solve the time independent
268
00:09:22,240 --> 00:09:28,399
schrodinger equation which by the way we
269
00:09:24,320 --> 00:09:28,399
sometimes call TISE
270
00:09:29,600 --> 00:09:34,800
then for free we get the time evolution
271
00:09:32,000 --> 00:09:34,800
of the wave function
272
00:09:35,600 --> 00:09:42,959
where phi(x) is the solution to the
273
00:09:39,279 --> 00:09:44,880
TISE so while this is in general
274
00:09:42,959 --> 00:09:46,880
impossible to solve there are a set of
275
00:09:44,880 --> 00:09:47,440
very important cases which is possible
276
00:09:46,880 --> 00:09:48,800
to solve
277
00:09:47,440 --> 00:09:50,800
and those are the ones that we'll be
278
00:09:48,800 --> 00:09:52,720
looking at in this course
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00:09:50,800 --> 00:09:54,560
okay thank you for your time it must have been
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00:09:52,720 --> 00:09:55,440
very hot in the 1920s if this suit is
281
00:09:54,560 --> 00:09:57,600
something to go by
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00:09:55,440 --> 00:10:00,080
I think I'll turn back into my normal
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00:09:57,600 --> 00:10:00,080
clothes
284
00:10:03,200 --> 00:10:08,880
excuse me Geoffrey sorry about that
285
00:10:06,560 --> 00:10:10,560
so i'll see you in the next video where
286
00:10:08,880 --> 00:10:12,480
we will take a look at some particular
287
00:10:10,560 --> 00:10:18,959
solutions to the schrodinger equation
288
00:10:12,480 --> 00:10:18,959
thank you
V1.3 Plane waves
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
plane wave solutions to the Schroedinger equation in the absence of a potential; compatibility with the Einstein relation E=?? and the de Broglie relation p=?k.
0:00:00.799,0:00:04.480
hello in this video we're going to take
0:00:03.120,0:00:06.160
a look at some
0:00:04.480,0:00:08.240
specific solutions to the schroedinger
0:00:06.160,0:00:12.080
equation so remember the
0:00:08.240,0:00:16.000
time independent schroedinger equation
0:00:12.080,0:00:16.800
is written in terms of the hamiltonian
0:00:16.000,0:00:20.240
acting
0:00:16.800,0:00:21.920
on some wave function phi(x) and this
0:00:20.240,0:00:23.039
is just the time independent part
0:00:21.920,0:00:24.160
remember we can always add the time
0:00:23.039,0:00:28.080
dependent part in
0:00:24.160,0:00:30.240
later this is defined to be
0:00:28.080,0:00:34.800
-hbar^2/2m . grad^2
0:00:30.240,0:00:38.079
plus V(x)
0:00:34.800,0:00:39.360
all acting on phi(x) and this thing
0:00:38.079,0:00:45.680
equals E phi(x)
0:00:42.960,0:00:47.200
so in one dimension which is the case
0:00:45.680,0:00:48.160
we'll be interested in in most of this
0:00:47.200,0:00:50.640
course
0:00:48.160,0:00:52.320
we can write this more simply by just
0:00:50.640,0:00:56.079
taking the more complicated power here
0:00:52.320,0:00:59.600
as -hbar^2/2m phi''(x)
0:00:56.079,0:01:03.280
plus
0:00:59.600,0:01:06.560
V(x) phi(x)
0:01:03.280,0:01:09.520
equals E phi(x)
0:01:06.560,0:01:11.200
so in this case it's just a
0:01:09.520,0:01:13.520
second order ordinary differential
0:01:11.200,0:01:15.280
equation
0:01:13.520,0:01:16.960
it's an eigenvalue problem we need to
0:01:15.280,0:01:20.880
find the eigenfunctions
0:01:16.960,0:01:22.720
phi(x) which solves this equation and the
0:01:20.880,0:01:23.920
corresponding eigenvalues E which will
0:01:22.720,0:01:25.360
be the energies of the
0:01:23.920,0:01:28.560
system the energies the particles can
0:01:25.360,0:01:29.360
take and in general to specify such a
0:01:28.560,0:01:30.880
problem
0:01:29.360,0:01:32.720
if we've got some physical system we
0:01:30.880,0:01:36.000
want to model with quantum mechanics
0:01:32.720,0:01:40.400
we just write down a potential
0:01:36.000,0:01:43.200
that encodes that system and then
0:01:40.400,0:01:44.560
we have to solve the time independent
0:01:43.200,0:01:46.479
Schrodinger equation
0:01:44.560,0:01:48.000
for our potential subject to boundary
0:01:46.479,0:01:51.119
conditions
0:01:48.000,0:01:52.320
so the simplest possible potential we
0:01:51.119,0:01:54.720
can consider
0:01:52.320,0:01:56.240
is just the case where the potential
0:01:54.720,0:01:59.200
is equal to zero
0:01:56.240,0:01:59.200
so the simplest case
0:02:02.159,0:02:05.759
V=0 and when the potential
0:02:04.719,0:02:07.360
is equal to zero
0:02:05.759,0:02:09.440
our time independent schrodinger
0:02:07.360,0:02:13.360
equation just reads
0:02:09.440,0:02:17.840
-hbar^2/2m phi''(x)=E phi(x)
0:02:13.360,0:02:21.360
this can be solved with
0:02:17.840,0:02:24.959
an ansatz so we can say that phi(x)
0:02:21.360,0:02:28.400
is equal to a plus or minus e to the i
0:02:24.959,0:02:28.400
plus or minus k x
0:02:29.040,0:02:33.440
where a plus or minus are just some
0:02:31.599,0:02:35.120
arbitrary coefficients
0:02:33.440,0:02:37.120
and when we substitute this in we
0:02:35.120,0:02:41.519
find that
0:02:37.120,0:02:45.360
hbar^2 k^2/2m = E
0:02:41.519,0:02:46.879
and in other words
0:02:45.360,0:02:48.480
the energy eigenvalues E that we've
0:02:46.879,0:02:53.120
solved for equal
0:02:48.480,0:02:54.800
hbar^2 k^2 / 2m
0:02:53.120,0:02:56.239
and now when we take a look back at the
0:02:54.800,0:02:57.519
problem we're trying to solve this
0:02:56.239,0:02:58.480
here's the time independent schrodinger
0:02:57.519,0:02:59.599
equation in 1D
0:02:58.480,0:03:01.840
we're trying to solve for the energy
0:02:59.599,0:03:03.280
eigenvalues. Energies of course have
0:03:01.840,0:03:05.680
two contributions they have
0:03:03.280,0:03:07.360
the potential energy term which is here
0:03:05.680,0:03:08.879
and the kinetic energy term which must
0:03:07.360,0:03:11.040
be this thing over here
0:03:08.879,0:03:14.159
so we've set the potential equal to zero
0:03:11.040,0:03:15.680
so the energy should be purely kinetic
0:03:14.159,0:03:17.760
and the kinetic energy we'd usually
0:03:15.680,0:03:21.040
expect to be able to write
0:03:17.760,0:03:24.720
p^2/2m
0:03:21.040,0:03:27.840
so this is true provided that
0:03:24.720,0:03:27.840
p = hbar k
0:03:28.000,0:03:33.840
you'll see from here that this is nothing
0:03:30.159,0:03:33.840
other than our de Broglie relation
0:03:37.280,0:03:44.080
which tells us that
0:03:41.040,0:03:47.200
all quantum particles have a
0:03:44.080,0:03:50.560
wave-like description as well and the
0:03:47.200,0:03:53.280
momentum p of a of the particle
0:03:50.560,0:03:54.959
corresponds to a wave vector k for the
0:03:53.280,0:03:57.040
wave
0:03:54.959,0:03:58.840
we can also write this as p equals h
0:03:57.040,0:04:00.480
over lambda where lambda is the
0:03:58.840,0:04:03.200
wavelength
0:04:00.480,0:04:03.519
so I said that in general solving the
0:04:03.200,0:04:04.799
time
0:04:03.519,0:04:06.000
independent schrodinger equation is
0:04:04.799,0:04:07.360
actually the tricky bit that we've
0:04:06.000,0:04:09.680
already done here.
0:04:07.360,0:04:10.799
We then get the time dependence
0:04:09.680,0:04:14.480
for free
0:04:10.799,0:04:19.840
so let's take a look at the time
0:04:14.480,0:04:19.840
dependent schrodinger equation TDSE
0:04:20.000,0:04:26.560
so this reads i h bar
0:04:23.120,0:04:28.240
psi dot (where the dot indicates
0:04:26.560,0:04:30.479
the partial derivative of psi with
0:04:28.240,0:04:34.160
respect to time holding position
0:04:30.479,0:04:36.720
constant) equals H psi
0:04:34.160,0:04:38.000
in general and in this case this
0:04:36.720,0:04:39.040
equals E psi because we've already
0:04:38.000,0:04:43.199
solved the time
0:04:39.040,0:04:44.479
independent part so psi(x,t)
0:04:43.199,0:04:46.240
adds in the time dependence
0:04:44.479,0:04:48.880
corresponding to phi(x)
0:04:46.240,0:04:50.720
that we've already solved for so we
0:04:48.880,0:04:54.000
can again solve this with an ansatz
0:04:50.720,0:04:58.080
let's say that psi(x,t)
0:04:54.000,0:05:01.120
is equal to A plus or minus e to the
0:04:58.080,0:05:04.400
i plus or minus k x
0:05:01.120,0:05:07.680
as before and this time minus omega t
0:05:04.400,0:05:09.440
so remember that the time dependence
0:05:07.680,0:05:12.560
of an energy eigenvalue always just adds
0:05:09.440,0:05:14.400
this phase winding term
0:05:12.560,0:05:15.600
when we substitute this in we bring down
0:05:14.400,0:05:18.479
a minus i
0:05:15.600,0:05:20.000
omega there's an i here already the i
0:05:18.479,0:05:20.479
cancels with the minus sign and we find
0:05:20.000,0:05:24.000
that
0:05:20.479,0:05:27.759
hbar omega psi = E psi
0:05:24.000,0:05:30.880
or in other words E equals
0:05:27.759,0:05:35.440
hbar omega
0:05:30.880,0:05:35.440
which is nothing other than our Einstein
0:05:36.840,0:05:43.199
relation
0:05:39.520,0:05:45.039
recall Einstein said:
0:05:43.199,0:05:46.800
take light which is classically
0:05:45.039,0:05:49.280
described by a a wave
0:05:46.800,0:05:50.880
and we can say that another way to
0:05:49.280,0:05:52.000
think of that is that it's made up of
0:05:50.880,0:05:55.600
individual packets of
0:05:52.000,0:05:58.080
energy called quanta or photons and
0:05:55.600,0:05:59.199
if a photon has angular frequency omega
0:05:58.080,0:06:02.639
it has energy
0:05:59.199,0:06:05.919
E another way to write this is
0:06:02.639,0:06:06.960
E = hf where f is the frequency
0:06:05.919,0:06:08.639
of the photon
0:06:06.960,0:06:09.520
now we're not describing photons with
0:06:08.639,0:06:10.479
the Schrodinger equation we're
0:06:09.520,0:06:12.400
describing
0:06:10.479,0:06:14.240
non-relativistic massive particles such
0:06:12.400,0:06:16.240
as electrons
0:06:14.240,0:06:17.280
but these two obey something like an
0:06:16.240,0:06:19.680
Einstein relation
0:06:17.280,0:06:21.199
and that's by construction part of the
0:06:19.680,0:06:24.400
Schroedinger equation
0:06:21.199,0:06:25.919
so that's the first example of a simple
0:06:24.400,0:06:29.759
solution to the Schroedinger equation
0:06:25.919,0:06:29.759
okay thank you for your time
V1.4 Amplitudes and probabilities
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
probability amplitudes, probability densities, probability density currents, the continuity equation and local conservation of probability, and general boundary conditions on the wave function.
0:00:00.880,0:00:05.920
hello we've seen previously
0:00:03.600,0:00:07.600
that in quantum mechanics we describe
0:00:05.920,0:00:09.360
particles in terms of a quantity called
0:00:07.600,0:00:11.599
the wave function denoted with the greek
0:00:09.360,0:00:12.960
letter psi
0:00:11.599,0:00:14.880
which tells us something about the
0:00:12.960,0:00:16.160
probability of finding the particle
0:00:14.880,0:00:18.320
but it can't tell us exactly the
0:00:16.160,0:00:19.760
probability because psi is in general a
0:00:18.320,0:00:22.160
complex number
0:00:19.760,0:00:24.160
additionally it has what's called a
0:00:22.160,0:00:25.199
global phase so it has some complex
0:00:24.160,0:00:27.039
phase out the front
0:00:25.199,0:00:28.720
which is gauge dependent meaning that
0:00:27.039,0:00:31.920
it's a mathematical choice
0:00:28.720,0:00:33.920
as to what value that phase takes. It's
0:00:31.920,0:00:36.880
a mathematical redundancy in the system
0:00:33.920,0:00:38.640
so psi itself can't tell us about
0:00:36.880,0:00:41.920
probabilities. In fact it denotes
0:00:38.640,0:00:43.440
what's called a probability amplitude
0:00:41.920,0:00:45.200
which we usually just refer to as the
0:00:43.440,0:00:47.039
amplitude
0:00:45.200,0:00:48.480
to identify the probability itself we
0:00:47.039,0:00:50.320
need to use what's called the
0:00:48.480,0:00:53.440
Born rule
0:00:50.320,0:00:56.239
which tells us the following
0:00:53.440,0:00:58.000
the square-modulus of psi multiplied by
0:00:56.239,0:00:59.920
a small line element dx
0:00:58.000,0:01:01.760
in one dimension gives the probability
0:00:59.920,0:01:05.040
to find the particle between x
0:01:01.760,0:01:07.520
and x+dx at time t. In general
0:01:05.040,0:01:09.200
this will be a small volume element in
0:01:07.520,0:01:11.200
three dimensions
0:01:09.200,0:01:12.880
so it's this quantity that we need to be
0:01:11.200,0:01:14.479
looking at for probabilities
0:01:12.880,0:01:17.600
and this has all the properties we'd
0:01:14.479,0:01:20.479
like: it's a real number
0:01:17.600,0:01:22.400
it no longer has this arbitrary global
0:01:20.479,0:01:24.799
phase out of the front
0:01:22.400,0:01:26.560
and in fact if we integrate the modulus
0:01:24.799,0:01:28.960
square of psi across
0:01:26.560,0:01:30.159
all of space we'll get the value one
0:01:28.960,0:01:31.200
because even though we don't know where
0:01:30.159,0:01:34.079
the particle is
0:01:31.200,0:01:35.280
we know that it must exist somewhere.
0:01:34.079,0:01:37.439
We can identify
0:01:35.280,0:01:40.000
this quantity here as what's called the
0:01:37.439,0:01:42.000
probability density.
0:01:40.000,0:01:44.000
This integrated over a region
0:01:42.000,0:01:46.240
of space gives the probability to find
0:01:44.000,0:01:48.240
the particle within that region
0:01:46.240,0:01:49.600
since integrating it across all of
0:01:48.240,0:01:51.200
space gives the value one
0:01:49.600,0:01:53.360
we actually have what's called the
0:01:51.200,0:01:55.360
'global conservation of probability'
0:01:53.360,0:01:56.880
the probability of finding a particle is
0:01:55.360,0:01:58.399
always a constant
0:01:56.880,0:02:00.079
in fact there's a stronger condition on
0:01:58.399,0:02:01.759
the probability which we'll take a look
0:02:00.079,0:02:03.520
at now with a worked example
0:02:01.759,0:02:09.840
let me just move over to my worked
0:02:03.520,0:02:09.840
example area
0:02:13.920,0:02:17.760
okay so we have the global conservation
0:02:15.840,0:02:19.120
of probability but to get a stronger
0:02:17.760,0:02:22.959
constraint on it
0:02:19.120,0:02:24.239
let's take a look at our probability
0:02:22.959,0:02:27.680
density and let's look at the time
0:02:24.239,0:02:27.680
derivative of it.
0:02:28.239,0:02:31.760
Our
0:02:30.560,0:02:36.000
probability density
0:02:31.760,0:02:39.200
is defined as |psi|^2
0:02:36.000,0:02:42.720
which is equal to psi*.psi
0:02:39.200,0:02:44.239
well I've omitted the
0:02:42.720,0:02:46.000
position and time dependence but they're
0:02:44.239,0:02:48.480
there
0:02:46.000,0:02:49.680
and so what we'd like is the time
0:02:48.480,0:02:51.360
derivative of this the partial
0:02:49.680,0:02:52.560
derivative of the probability density
0:02:51.360,0:02:56.160
with respect to time
0:02:52.560,0:02:57.440
holding position constant and so this
0:02:56.160,0:03:00.959
of course will be equal to
0:02:57.440,0:03:07.360
d(psi*)/dt.psi + psi*.d(psi)/dt
0:03:04.239,0:03:08.879
from the chain rule and then to
0:03:07.360,0:03:10.400
work out what these quantities are
0:03:08.879,0:03:13.440
we can use the time dependent Schrodinger
0:03:10.400,0:03:17.280
equation which tells us that
0:03:17.280,0:03:23.440
hbar psi dot equals
0:03:21.040,0:03:24.159
(I'll write the hamiltonian out in full)
0:03:23.440,0:03:30.480
(-hbar^2/2m grad^2)psi
0:03:27.360,0:03:33.840
plus
0:03:30.480,0:03:35.840
V psi and so
0:03:33.840,0:03:37.760
dividing by hbar and
0:03:35.840,0:03:41.599
multiplying by minus i
0:03:37.760,0:03:45.200
we have d(psi)/dt equals
0:03:41.599,0:03:49.440
minus i over h bar
0:03:45.200,0:03:52.560
-hbar^2/2m grad^2 psi
0:03:49.440,0:03:56.400
+ V psi
0:03:52.560,0:03:59.680
how about d(psi*)/dt
0:03:56.400,0:04:01.360
we just take the
0:03:59.680,0:04:05.200
complex conjugate of this
0:04:01.360,0:04:08.640
and so we get d(psi*)/dt
0:04:05.200,0:04:12.159
is equal to
0:04:08.640,0:04:15.760
minus hbar^2/2m grad^2 psi
0:04:12.159,0:04:19.120
plus
0:04:15.760,0:04:22.720
V psi*
0:04:19.120,0:04:24.639
where the potential is assumed real
0:04:22.720,0:04:26.800
okay so substituting both of these back
0:04:24.639,0:04:30.560
into this expression up here
0:04:26.800,0:04:33.919
we find that we have d(rho)/dt
0:04:30.560,0:04:37.040
is equal to so we get the i
0:04:33.919,0:04:40.080
over h bar out the front
0:04:37.040,0:04:50.080
-hbar^2/2m grad^2 psi* + v psi*
0:04:46.479,0:04:54.080
and all of this
0:04:50.080,0:04:57.280
gets multiplied by psi plus
0:04:54.080,0:05:00.479
well so sorry it would be uh
0:04:57.280,0:05:03.199
plus psi* times psi
0:05:00.479,0:05:05.039
dot psi yes psi dot but remember
0:05:03.199,0:05:06.720
that there was a minus sign in front of
0:05:05.039,0:05:08.800
the psi dot compared to the
0:05:06.720,0:05:12.160
d(psi*)/dt
0:05:08.800,0:05:12.160
actually this should have been a minus
0:05:12.240,0:05:16.560
and then otherwise inside it's pretty
0:05:14.240,0:05:20.560
much the same thing
0:05:16.560,0:05:20.560
with psi instead of psi*
0:05:22.080,0:05:29.280
okay so here we have V psi* psi
0:05:25.520,0:05:32.560
here we have minus V psi* psi
0:05:29.280,0:05:35.919
and so these are the same thing
0:05:32.560,0:05:38.479
so what's left
0:05:35.919,0:05:40.560
we have -hbar^2/2m
0:05:38.479,0:05:42.639
out the front of both expressions
0:05:40.560,0:05:44.720
and we can bring that out the front to
0:05:42.639,0:05:48.400
get d(rho)/dt
0:05:44.720,0:05:55.759
= - i hbar/2m
0:05:53.360,0:05:57.360
let's bring the psi the left of
0:05:55.759,0:05:59.039
grad^2 psi* because
0:05:57.360,0:06:01.520
the grad^2 only acts on the psi*
0:05:59.039,0:06:04.840
and inside these parentheses
0:06:01.520,0:06:07.680
but psi grad^2 psi*
0:06:04.840,0:06:15.600
minus psi* grad^2 psi
0:06:12.800,0:06:16.080
okay to go a bit further we need to
0:06:15.600,0:06:19.120
use
0:06:16.080,0:06:20.639
an identity from vector calculus
0:06:19.120,0:06:22.160
I'm just going to fold this paper over
0:06:20.639,0:06:25.039
just
0:06:22.160,0:06:25.039
a bit more closely
0:06:26.319,0:06:32.319
so we know that
0:06:30.880,0:06:34.240
and you can just you can derive this
0:06:32.319,0:06:35.520
yourself or look it up but we have
0:06:34.240,0:06:37.280
I'm going to look it up off my bit of
0:06:35.520,0:06:40.720
paper so if we have
0:06:37.280,0:06:41.360
f grad^2 g we can always write
0:06:40.720,0:06:44.639
this
0:06:41.360,0:06:48.479
as the divergence of f grad g
0:06:44.639,0:06:52.000
minus
0:06:48.479,0:06:55.440
(grad f).(grad g)
0:06:52.000,0:06:59.039
so
0:06:55.440,0:07:01.360
taking a look again at our
0:06:59.039,0:07:02.400
previous expression here in the first
0:07:01.360,0:07:06.880
term
0:07:02.400,0:07:06.880
f is psi and g is psi*
0:07:07.120,0:07:13.360
and in the second term those two switch
0:07:10.560,0:07:14.160
and so we end up with the result that
0:07:13.360,0:07:17.919
d(rho)/dt
0:07:14.160,0:07:23.039
equals
0:07:17.919,0:07:23.039
-i hbar/2m
0:07:24.400,0:07:30.960
divergence of i grad
0:07:27.440,0:07:34.240
psi*
0:07:30.960,0:07:38.240
minus (grad psi).(grad psi*)
0:07:38.479,0:07:45.840
minus the divergence of (switch all
0:07:41.840,0:07:45.840
psi and psi*)
0:07:52.639,0:07:59.120
like so and we see that this term is the
0:07:56.240,0:08:01.520
same as this term so these cancel
0:07:59.120,0:08:02.479
and the rest of the terms that remain
0:08:01.520,0:08:04.960
have a
0:08:02.479,0:08:06.319
divergence term on the outside and so we
0:08:04.960,0:08:10.560
can say that
0:08:06.319,0:08:13.199
d(rho)/dt is equal to minus
0:08:10.560,0:08:15.440
the divergence of some quantity we call
0:08:13.199,0:08:15.440
j
0:08:17.360,0:08:24.160
where j, which is in general
0:08:20.720,0:08:26.160
a function of position and time, is equal
0:08:24.160,0:08:27.680
to:
0:08:26.160,0:08:29.919
we've taken the minus out the front so:
0:08:41.839,0:08:45.200
and it's a vector quantity in three
0:08:44.159,0:08:49.120
dimensions
0:08:45.200,0:08:49.120
because of the grad here
0:08:51.120,0:08:55.680
so this is our stronger constraint it
0:08:54.320,0:08:57.839
tells us that not only is
0:08:55.680,0:08:59.680
probability conserved globally but it's
0:08:57.839,0:09:00.880
actually conserved locally
0:08:59.680,0:09:03.040
and what this means is that this
0:09:00.880,0:09:04.320
expression here is an expression for a
0:09:03.040,0:09:09.839
general fluid which
0:09:04.320,0:09:09.839
is what's called a continuity equation
0:09:12.959,0:09:17.200
and it tells us that imagine we have
0:09:15.279,0:09:18.640
some little box in space
0:09:17.200,0:09:21.040
which originally had some probability
0:09:18.640,0:09:22.080
density but this goes down it changes in
0:09:21.040,0:09:24.480
time
0:09:22.080,0:09:26.240
well it would be consistent with global
0:09:24.480,0:09:29.360
conservation of probability
0:09:26.240,0:09:30.480
to have another little box separated off
0:09:29.360,0:09:31.920
in space somewhere else
0:09:30.480,0:09:34.320
have the probability go up just
0:09:31.920,0:09:35.839
instantaneously that would be
0:09:34.320,0:09:37.600
compatible with global conservation of
0:09:35.839,0:09:39.360
probability but
0:09:37.600,0:09:41.680
what the continuity equation tells us is
0:09:39.360,0:09:43.600
that actually it's locally conserved
0:09:41.680,0:09:45.440
and so there must be a flow of
0:09:43.600,0:09:46.160
probability density between these two
0:09:45.440,0:09:47.920
points
0:09:46.160,0:09:49.360
that is, if the probability densities go
0:09:47.920,0:09:50.399
down in one region there must be a
0:09:49.360,0:09:52.880
divergence
0:09:50.399,0:09:54.560
of this current out of that region there
0:09:52.880,0:09:56.399
must be some flow out of this region
0:09:54.560,0:09:58.160
into that region
0:09:56.399,0:09:59.920
and so for it to go down from this box
0:09:58.160,0:10:01.760
and up in this box there must be a flow
0:09:59.920,0:10:05.839
between the two boxes
0:10:01.760,0:10:06.720
and this flow is a flow of this
0:10:05.839,0:10:10.399
quantity j(x,t)
0:10:06.720,0:10:10.399
which is called the probability
0:10:11.600,0:10:19.839
current density
0:10:21.040,0:10:24.800
there we go probability current density
0:10:23.519,0:10:27.519
so let's
0:10:24.800,0:10:29.760
take a look at some general boundary
0:10:27.519,0:10:33.070
conditions that we can apply on psi
0:10:29.760,0:10:36.299
as a result of these ideas
0:10:38.240,0:10:41.120
let's take a look at some boundary
0:10:39.279,0:10:42.560
conditions then the first boundary
0:10:41.120,0:10:43.680
condition on the wave function is that
0:10:42.560,0:10:46.800
it has to be continuous
0:10:43.680,0:10:49.519
across space
0:10:46.800,0:10:50.800
and the second condition is that the
0:10:49.519,0:10:56.079
first derivative of
0:10:50.800,0:10:56.079
psi with respect to x is also continuous
0:10:56.880,0:11:01.120
in fact the second condition doesn't
0:10:59.600,0:11:04.079
hold in all cases
0:11:01.120,0:11:05.440
but they are fairly pathological
0:11:04.079,0:11:08.640
cases where it doesn't hold
0:11:05.440,0:11:10.160
in fact psi prime the first derivative
0:11:08.640,0:11:12.800
with respect to position
0:11:10.160,0:11:14.560
is continuous provided that there's
0:11:12.800,0:11:15.279
no infinite discontinuity in the
0:11:14.560,0:11:16.320
potential
0:11:15.279,0:11:18.399
at places where there's infinite
0:11:16.320,0:11:19.279
discontinuities this may not hold so
0:11:18.399,0:11:21.839
let's add that
0:11:19.279,0:11:21.839
caveat
0:11:22.320,0:11:26.160
so these boundary conditions really come
0:11:24.320,0:11:27.600
from the fact that
0:11:26.160,0:11:29.920
we don't consider potentials which are
0:11:27.600,0:11:32.800
too pathological in fact we can
0:11:29.920,0:11:34.720
consider jumps in our potential we
0:11:32.800,0:11:36.079
can even have delta function potentials
0:11:34.720,0:11:38.560
but we don't consider anything worse
0:11:36.079,0:11:38.560
than that
0:11:38.640,0:11:41.760
a final condition which is not really a
0:11:40.079,0:11:42.560
boundary condition but which generally
0:11:41.760,0:11:44.480
applies
0:11:42.560,0:11:46.640
is that in regions where the potential
0:11:44.480,0:11:49.440
is infinite we require the wave function
0:11:46.640,0:11:49.440
to go to zero
0:11:49.680,0:11:52.880
the reason for this is that the modulus
0:11:51.519,0:11:54.399
square of the wave function is the
0:11:52.880,0:11:56.240
probability density
0:11:54.399,0:11:57.839
and we require the probability density
0:11:56.240,0:11:59.040
to be zero in regions where the
0:11:57.839,0:12:00.959
potential is infinity
0:11:59.040,0:12:03.040
in order to keep the energy finite which
0:12:00.959,0:12:06.320
is something we'd like to do
0:12:03.040,0:12:07.440
okay so let's take a look at some of the
0:12:06.320,0:12:10.639
philosophical
0:12:07.440,0:12:13.120
ideas behind the interpretation of these
0:12:10.639,0:12:13.120
objects
0:12:18.560,0:12:22.079
Dirac in his textbook gives a very good
0:12:20.480,0:12:23.519
explanation as to why we might expect
0:12:22.079,0:12:24.560
probabilities to come up in quantum
0:12:23.519,0:12:27.360
mechanics
0:12:24.560,0:12:27.920
we'd like descriptions of things such as
0:12:27.360,0:12:29.440
light
0:12:27.920,0:12:31.600
which are familiar on the macroscopic
0:12:29.440,0:12:33.600
scale but descriptions which
0:12:31.600,0:12:36.240
work down on the microscopic scale
0:12:33.600,0:12:37.680
when we see a beam of light like the
0:12:36.240,0:12:41.279
spot from this laser
0:12:37.680,0:12:43.440
pen we can of course
0:12:41.279,0:12:45.680
describe this classically and we can use
0:12:43.440,0:12:49.200
Maxwell's equations to describe it
0:12:45.680,0:12:52.160
but we'd like a quantum description
0:12:49.200,0:12:52.880
so Dirac gives us the example of taking
0:12:52.160,0:12:55.120
a crystal
0:12:52.880,0:12:57.360
something like calcite which I have here
0:12:55.120,0:12:59.519
so calcite is birefringent
0:12:57.360,0:13:00.720
which means if I shine the laser through
0:12:59.519,0:13:03.120
the calcite
0:13:00.720,0:13:03.839
my one spot should turn into two you can
0:13:03.120,0:13:05.680
see there
0:13:03.839,0:13:07.519
and as I rotate the crystal I actually
0:13:05.680,0:13:09.120
want to rotate around the other
0:13:07.519,0:13:11.040
and these two spots have different
0:13:09.120,0:13:14.000
polarizations the key thing is just that
0:13:11.040,0:13:14.000
there's two spots now
0:13:14.959,0:13:19.440
so why is this a problem well it's not a
0:13:17.920,0:13:20.959
problem on the classical scale
0:13:19.440,0:13:22.959
because we just say well there's a beam
0:13:20.959,0:13:24.000
of light it has some amplitude and some
0:13:22.959,0:13:26.959
intensity
0:13:24.000,0:13:28.079
and the intensity just splits into those
0:13:26.959,0:13:29.600
two beams
0:13:28.079,0:13:31.440
but if we're going to come up with any
0:13:29.600,0:13:32.959
kind of description of this on the scale
0:13:31.440,0:13:34.399
of single particles
0:13:32.959,0:13:36.480
it's going to have to somehow account
0:13:34.399,0:13:37.839
for each particle either going
0:13:36.480,0:13:39.600
one way or the other so there's got to
0:13:37.839,0:13:41.680
be some kind of probabilistic element to
0:13:39.600,0:13:44.639
it
0:13:41.680,0:13:46.000
the same idea leads to problems in the
0:13:44.639,0:13:48.720
idea of the atom
0:13:46.000,0:13:49.839
so we already had a problem after the
0:13:48.720,0:13:52.320
Rutherford experiment
0:13:49.839,0:13:54.399
showed that atoms have a positive
0:13:52.320,0:13:55.760
nucleus with negative electric charge
0:13:54.399,0:13:57.199
around it why doesn't the negative
0:13:55.760,0:13:59.040
charge drop into the positive and
0:13:57.199,0:14:00.720
minimize its energy
0:13:59.040,0:14:02.240
when Schroedinger wrote down his
0:14:00.720,0:14:04.240
description of the atom according to the
0:14:02.240,0:14:06.880
Schroedinger equation
0:14:04.240,0:14:08.560
he initially had the idea that the
0:14:06.880,0:14:10.639
probability current density that we've
0:14:08.560,0:14:13.040
just identified
0:14:10.639,0:14:14.240
may describe the current density of
0:14:13.040,0:14:16.320
the electron
0:14:14.240,0:14:17.680
as it orbits the nucleus but then that
0:14:16.320,0:14:20.639
gives us a problem again
0:14:17.680,0:14:22.399
because if the electron is really
0:14:20.639,0:14:24.160
orbiting the nucleus then
0:14:22.399,0:14:25.360
an electric charge
0:14:24.160,0:14:27.920
moving in a circle
0:14:25.360,0:14:30.240
radiates energy and so it should lose
0:14:27.920,0:14:32.079
its energy and drop into nucleus again
0:14:30.240,0:14:33.360
so according to Feynman in the Feynman
0:14:32.079,0:14:34.880
lectures, Schroedinger originally
0:14:33.360,0:14:37.040
interpreted the probability current
0:14:34.880,0:14:38.160
density as a literal current density of
0:14:37.040,0:14:40.000
the electron
0:14:38.160,0:14:42.160
the idea was that when we go down to the
0:14:40.000,0:14:44.320
quantum scale and we look at
0:14:42.160,0:14:46.560
individual particles such as photons in
0:14:44.320,0:14:48.560
this case each photon will be kind of
0:14:46.560,0:14:51.279
spread out in exactly the same way that
0:14:48.560,0:14:54.399
the intensity of the light is spread
0:14:51.279,0:14:55.920
but that can't be true because when we
0:14:54.399,0:14:57.680
observe the particles we always observe
0:14:55.920,0:14:58.959
them at a single location
0:14:57.680,0:15:00.399
it also can't be true because if we
0:14:58.959,0:15:02.399
think about the electron trying to orbit
0:15:00.399,0:15:03.920
the nucleus it would be losing energy
0:15:02.399,0:15:05.360
this is where Born came along with the
0:15:03.920,0:15:06.560
Born rule and why the Born rule was so
0:15:05.360,0:15:08.959
important
0:15:06.560,0:15:10.560
he told us that the current density was
0:15:08.959,0:15:11.600
not the current density of a smeared out
0:15:10.560,0:15:14.079
electron
0:15:11.600,0:15:15.600
it's the current density of a flow of
0:15:14.079,0:15:17.040
probability but when you look for the
0:15:15.600,0:15:17.920
electron you always find it in one
0:15:17.040,0:15:20.000
specific place
0:15:17.920,0:15:21.199
the electron is not spread out quantum
0:15:20.000,0:15:22.639
particles aren't spread out their
0:15:21.199,0:15:24.079
probabilities are spread out
0:15:22.639,0:15:26.399
according to their amplitudes described
0:15:24.079,0:15:27.519
by the wave function and Born told us
0:15:26.399,0:15:29.920
that the modulus square of that
0:15:27.519,0:15:30.959
amplitude gives us the probability of
0:15:29.920,0:15:33.600
finding the particle
0:15:30.959,0:15:36.240
in a little region okay thanks for your
0:15:33.600,0:15:36.240
time
V1.5 Two slit demo
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
a demonstration of Young's two slit experiment.
0:00:00.719,0:00:03.840
hello in this video i'm going to give a
0:00:02.720,0:00:05.600
quick demonstration
0:00:03.840,0:00:07.359
as to how to do the two-slit experiment
0:00:05.600,0:00:10.480
in your own room
0:00:07.359,0:00:12.160
so we're going to need two narrow
0:00:10.480,0:00:14.320
closely spaced slits
0:00:12.160,0:00:15.679
through which we pass light
0:00:14.320,0:00:16.480
the light can interfere through the two
0:00:15.679,0:00:18.400
different slits
0:00:16.480,0:00:20.000
and hit some screen where we can measure
0:00:18.400,0:00:22.240
the interference pattern
0:00:20.000,0:00:23.920
and therefore deduce properties of wave
0:00:22.240,0:00:26.160
superposition and so on
0:00:23.920,0:00:28.160
so for the screen that we're going to
0:00:26.160,0:00:29.840
hit into to measure the pattern we can
0:00:28.160,0:00:32.640
just use the wall that'll be fine so
0:00:29.840,0:00:34.160
there's a webcam focused on the wall
0:00:32.640,0:00:35.440
for the divider between the two
0:00:34.160,0:00:37.600
slits that's the tricky bit because we
0:00:35.440,0:00:39.440
need something extremely thin
0:00:37.600,0:00:41.200
but we have lots of extremely thin
0:00:39.440,0:00:42.239
things coming out of our heads: our
0:00:41.200,0:00:45.600
hairs
0:00:42.239,0:00:47.920
so what I have here is a bit of
0:00:45.600,0:00:49.440
card where I've taped a hair across it
0:00:47.920,0:00:50.800
there's a slit I've cut
0:00:49.440,0:00:52.480
with some scissors but that's just to
0:00:50.800,0:00:54.399
cut the amount of light passing through
0:00:52.480,0:00:56.800
down so it's not too intense
0:00:54.399,0:00:57.600
and stuck vertically across that you
0:00:56.800,0:00:58.960
can see the bit of
0:00:57.600,0:01:00.800
duct tape where I've used it to tape
0:00:58.960,0:01:04.239
across here
0:01:00.800,0:01:04.239
vertically like this
0:01:04.320,0:01:10.400
so if I shine the laser through
0:01:07.760,0:01:10.400
through that
0:01:10.880,0:01:15.360
slit I'll get so I'm just going to use
0:01:13.520,0:01:18.400
this green laser here which
0:01:15.360,0:01:20.240
has a wavelength of 568 nanometers
0:01:18.400,0:01:22.240
it's just quite nice and bright so it's
0:01:20.240,0:01:24.400
convenient for this
0:01:22.240,0:01:25.600
so you can see the colour there okay so
0:01:24.400,0:01:28.960
if I shine it
0:01:25.600,0:01:32.159
through the slit
0:01:28.960,0:01:34.720
at a location where
0:01:32.159,0:01:36.159
there's no hair you'll just see a
0:01:34.720,0:01:38.479
spot on the wall
0:01:36.159,0:01:39.439
I've turned the exposure right down on
0:01:38.479,0:01:40.960
the camera
0:01:39.439,0:01:42.880
but there you see when it hits the hair
0:01:40.960,0:01:46.159
you see that pattern picks up
0:01:42.880,0:01:46.159
there we go just there
0:01:48.159,0:01:53.119
I'm using a pack of cards to get the height
0:01:49.759,0:01:55.840
just right to line up with the slit
0:01:53.119,0:01:55.840
there we go
0:02:00.640,0:02:05.360
there we go and it's convenient to
0:02:03.600,0:02:06.799
remove that central spot which isn't really
0:02:05.360,0:02:08.560
contributing to the pattern
0:02:06.799,0:02:10.479
to do that you can use something like
0:02:08.560,0:02:12.319
this: this is just a block of ink for
0:02:10.479,0:02:13.280
calligraphy but something dark and quite
0:02:12.319,0:02:14.959
thin
0:02:13.280,0:02:17.440
and we can place that in the way of the
0:02:14.959,0:02:18.879
central spot
0:02:17.440,0:02:23.840
and then if I just turn it back onto the
0:02:18.879,0:02:23.840
hair again there we go
0:02:26.640,0:02:30.720
so now we get the pattern without the
0:02:28.400,0:02:33.920
central
0:02:30.720,0:02:36.160
spot and actually you can
0:02:33.920,0:02:37.760
measure, if you know the distance to
0:02:36.160,0:02:39.760
the wall with the ruler
0:02:37.760,0:02:40.879
and you can measure the spacing between
0:02:39.760,0:02:42.800
these
0:02:40.879,0:02:44.800
different peaks which you can see here
0:02:42.800,0:02:47.120
which I estimate to be something like
0:02:44.800,0:02:49.519
around four millimeters a distance to the wall
0:02:47.120,0:02:51.920
of about 40 centimeters
0:02:49.519,0:02:52.879
you can use use this to deduce the
0:02:51.920,0:02:54.480
width of the hair
0:02:52.879,0:02:56.160
and I think mine's probably about
0:02:54.480,0:03:00.480
something like 100 times the
0:02:56.160,0:03:02.720
the wavelength of the light
0:03:00.480,0:03:04.480
I estimate and so that would make it
0:03:02.720,0:03:08.239
about
0:03:04.480,0:03:10.400
50 or so
0:03:08.239,0:03:11.680
micrometers with the width of this hair
0:03:10.400,0:03:12.800
and that seems to tally with what we
0:03:11.680,0:03:17.519
expect
0:03:12.800,0:03:17.519
okay thank you for your time
V2.1a Scattering from a potential step (part I)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
the forms of solutions to the TISE in regions of constant potential. Continued in video V2.1b.
0:00:02.399,0:00:04.720
Hello
0:00:03.360,0:00:07.040
in this video we're going to take a look
0:00:04.720,0:00:08.320
at solving the time independent
0:00:07.040,0:00:11.519
Schroedinger equation
0:00:08.320,0:00:12.960
in cases where we are scattering from a
0:00:11.519,0:00:15.280
potential step
0:00:12.960,0:00:16.320
so this is a particular instance of
0:00:15.280,0:00:17.760
solving the time independent
0:00:16.320,0:00:20.400
Schroedinger equation
0:00:17.760,0:00:23.359
in a case with a constant potential so
0:00:20.400,0:00:23.359
let's write that down
0:00:23.600,0:00:27.199
so our potential is going to be set
0:00:25.359,0:00:29.039
equal to V_0
0:00:27.199,0:00:31.840
and the form of the equation then looks
0:00:29.039,0:00:31.840
like this in one dimension
0:00:32.079,0:00:36.480
and the solutions to the equation in
0:00:34.960,0:00:39.120
general take different forms
0:00:36.480,0:00:40.000
depending on whether E the energy of the
0:00:39.120,0:00:42.640
particle
0:00:40.000,0:00:43.200
is greater than or less than V_0
0:00:42.640,0:00:45.520
the
0:00:43.200,0:00:47.600
constant potential in the problem so
0:00:45.520,0:00:52.480
first the case that E > V_0
0:00:48.559,0:00:52.480
we have plane wave solutions
0:00:53.600,0:00:57.360
so they take this form in general and
0:00:56.320,0:00:58.000
you can check quite easily by
0:00:57.360,0:01:00.640
substituting
0:00:58.000,0:01:01.199
those into here that these will solve
0:01:00.640,0:01:03.840
the
0:01:01.199,0:01:04.479
Schroedinger equation for particular
0:01:03.840,0:01:07.760
values
0:01:04.479,0:01:10.640
of k which we'll check in a second
0:01:07.760,0:01:13.600
the second case again for E > V_0
0:01:11.520,0:01:15.119
but well these correspond to
0:01:13.600,0:01:16.159
travelling waves when we add in the time
0:01:15.119,0:01:17.600
dependence
0:01:16.159,0:01:19.119
in the way that we always can add back
0:01:17.600,0:01:19.680
in the time dependence in terms of the
0:01:19.119,0:01:23.600
constant
0:01:19.680,0:01:23.600
phase winding
0:01:24.320,0:01:27.840
as a function of time these are two
0:01:26.799,0:01:30.240
traveling waves
0:01:27.840,0:01:31.759
but in certain cases depending on the
0:01:30.240,0:01:34.560
boundary conditions it can still be
0:01:31.759,0:01:36.240
important to look at standing waves
0:01:34.560,0:01:38.159
much like we can look at solutions to
0:01:36.240,0:01:39.920
the wave equation for sound say
0:01:38.159,0:01:42.320
we can have travelling waves of sound but
0:01:39.920,0:01:45.040
we can also have standing waves of sound
0:01:42.320,0:01:45.920
when trapped between two walls for
0:01:45.040,0:01:48.079
example
0:01:45.920,0:01:49.759
so in another case for E > V_0
0:01:48.079,0:01:51.920
we can have
0:01:49.759,0:01:55.520
well let's just write this down these
0:01:51.920,0:01:59.040
are travelling waves or plane waves
0:01:55.520,0:01:59.040
and we can also have standing waves
0:01:59.200,0:02:02.960
where of course you can in fact make
0:02:01.200,0:02:04.560
this form of solution
0:02:02.960,0:02:06.079
in terms of cosines and sines
0:02:04.560,0:02:08.000
from this solution in terms of
0:02:06.079,0:02:09.360
exponentials but this is going to be a
0:02:08.000,0:02:11.360
particularly convenient form
0:02:09.360,0:02:12.879
to consider in those cases where
0:02:11.360,0:02:14.480
standing waves are
0:02:12.879,0:02:15.840
the relevant things to consider let me
0:02:14.480,0:02:18.000
just magic my globe away for a second
0:02:15.840,0:02:19.920
with a quick slap of this board
0:02:18.000,0:02:22.160
good so let's write down that these are
0:02:19.920,0:02:24.480
standing waves
0:02:22.160,0:02:26.319
good now in the case where
0:02:24.480,0:02:27.920
E < V_0
0:02:26.319,0:02:29.440
the solutions are going to be
0:02:27.920,0:02:31.680
exponentially decaying
0:02:29.440,0:02:33.280
instead of being either travelling or
0:02:31.680,0:02:36.800
standing
0:02:33.280,0:02:39.920
so they take the following form where
0:02:36.800,0:02:43.120
kappa here has taken the place of k
0:02:39.920,0:02:44.080
and this is so that both k in
0:02:43.120,0:02:45.760
this case and
0:02:44.080,0:02:47.760
sorry k in this case and kappa in this
0:02:45.760,0:02:51.519
case are real numbers
0:02:47.760,0:02:52.319
so this looks like it doesn't
0:02:51.519,0:02:54.640
have an analogue
0:02:52.319,0:02:55.599
in terms of classical waves we have
0:02:54.640,0:02:58.319
both of these
0:02:55.599,0:02:59.200
in terms of sound and light this
0:02:58.319,0:03:00.720
looks like
0:02:59.200,0:03:02.400
there shouldn't be an analogue to just
0:03:00.720,0:03:05.519
something like a wave that just
0:03:02.400,0:03:07.280
exponentially grows or or decreases
0:03:05.519,0:03:08.800
but actually they do exist
0:03:07.280,0:03:11.840
classically and they're what's called
0:03:08.800,0:03:11.840
evanescent waves
0:03:12.000,0:03:15.280
or you might prefer to call them
0:03:12.959,0:03:17.519
evanescent solutions to the
0:03:15.280,0:03:18.560
equation because they're rather
0:03:17.519,0:03:19.120
strange to try and think of them as
0:03:18.560,0:03:21.360
waves.
0:03:19.120,0:03:24.560
I'll give you some examples of those
0:03:21.360,0:03:24.560
in a classical context separately.
0:03:24.879,0:03:28.799
So these are the general types of
0:03:26.879,0:03:31.760
solution we're going to be looking for
0:03:28.799,0:03:32.159
depending on how our energy relates to
0:03:31.760,0:03:35.120
our
0:03:32.159,0:03:36.159
potential so let's set up a particular
0:03:35.120,0:03:37.840
problem of interest
0:03:36.159,0:03:40.159
which is scattering from a potential
0:03:37.840,0:03:47.120
step so we'll clear this off and
0:03:40.159,0:03:47.120
specify the problem
V2.1b Scattering from a potential step (part II)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continuing from video V2.1a) scattering from a potential step in the case that the energy of the particle is greater than that of the step; setting up the problem, and solving for the probability amplitudes for reflection and transmission. Continued in video V2.1c.
0:00:04.480,0:00:06.399
good
0:00:05.120,0:00:08.320
okay so the potential we're going to
0:00:06.399,0:00:10.000
consider takes the following form
0:00:08.320,0:00:11.679
it's constant it has two
0:00:10.000,0:00:14.160
different constant values in different
0:00:11.679,0:00:14.160
regions
0:00:14.240,0:00:20.480
okay and let's draw it
0:00:17.359,0:00:22.720
okay so we're going to consider
0:00:20.480,0:00:23.600
a plane wave incident from the left from
0:00:22.720,0:00:26.000
over here
0:00:23.600,0:00:27.359
and we'll consider the cases E the
0:00:26.000,0:00:28.560
energy of that plane wave is greater
0:00:27.359,0:00:31.119
than V_0 first
0:00:28.560,0:00:31.760
and then E < V_0
0:00:31.119,0:00:33.840
second
0:00:31.760,0:00:37.120
and so we can think of this V_0 as
0:00:33.840,0:00:37.120
an actual potential step
0:00:38.160,0:00:40.800
so this region here as we'll see is
0:00:39.520,0:00:42.079
what's called the classically forbidden
0:00:40.800,0:00:43.360
region before we get to that
0:00:42.079,0:00:44.800
let's look at what happens to a plain
0:00:43.360,0:00:46.160
wave trying to travel over the top of
0:00:44.800,0:00:49.200
this
0:00:46.160,0:00:52.160
to do so let's head over to the worked
0:00:49.200,0:00:52.160
example area
0:00:57.600,0:01:04.479
okay so let's get the
0:01:00.879,0:01:07.200
board reproduced up there all right
0:01:04.479,0:01:09.280
so we can label the different regions
0:01:07.200,0:01:12.479
region one and region two that's called
0:01:09.280,0:01:15.520
region one this one over here
0:01:12.479,0:01:16.880
where x < 0 and
0:01:15.520,0:01:19.920
region two
0:01:16.880,0:01:23.360
this one up here where
0:01:19.920,0:01:25.520
x > 0
0:01:23.360,0:01:26.880
now we're going to send a plane wave in
0:01:25.520,0:01:29.439
from the left
0:01:26.880,0:01:30.000
so that's going to take the following
0:01:29.439,0:01:34.000
form so
0:01:30.000,0:01:34.000
let's animate that as it comes in
0:01:38.320,0:01:43.360
and it's going to be in region one
0:01:48.000,0:01:56.320
and we'll have
0:01:52.720,0:01:57.360
phi_in(x) equals...
0:01:56.320,0:01:59.119
okay so there's going to be a constant
0:01:57.360,0:01:59.680
pre-factor in general but it'll take the
0:01:59.119,0:02:02.880
form
0:01:59.680,0:02:05.200
e to the i k x
0:02:02.880,0:02:08.160
which is the form of a right going
0:02:05.200,0:02:09.840
travelling wave
0:02:08.160,0:02:11.360
remember we're always going to when
0:02:09.840,0:02:13.200
we put the time dependence back in this
0:02:11.360,0:02:15.680
will always multiply by
0:02:13.200,0:02:17.520
(and let's just put this in brackets here)
0:02:15.680,0:02:20.560
e to the minus i omega t
0:02:21.120,0:02:24.800
and kx minus omega t is the form of a
0:02:23.920,0:02:28.080
right travelling
0:02:24.800,0:02:29.760
plane wave okay so we can safely ignore
0:02:28.080,0:02:30.720
this bit for now
0:02:29.760,0:02:33.519
we'll just deal with the time
0:02:30.720,0:02:35.599
independent part so
0:02:33.519,0:02:37.280
phi_in is going to take the form
0:02:35.599,0:02:40.319
e to the ikx
0:02:37.280,0:02:42.319
in region 1 there will be some reflected
0:02:40.319,0:02:44.800
wave
0:02:42.319,0:02:45.360
so it's still in region 1 and we can
0:02:44.800,0:02:48.959
call it
0:02:45.360,0:02:50.160
phi_reflected as a function of x
0:02:48.959,0:02:52.160
it can potentially have a different
0:02:50.160,0:02:54.239
amplitude
0:02:52.160,0:02:55.200
at the front and it's going to take the
0:02:54.239,0:02:58.319
form e to the
0:02:55.200,0:03:00.319
sorry minus i k x because
0:02:58.319,0:03:01.920
that's going to be a left travelling wave
0:03:00.319,0:03:05.680
which you can see up here hopefully so
0:03:01.920,0:03:07.200
it'll reflect back
0:03:05.680,0:03:10.239
and it'll take the form of a
0:03:07.200,0:03:14.319
left-going traveling wave
0:03:10.239,0:03:14.319
in region 2 then
0:03:16.640,0:03:20.400
we will only have a right going wave
0:03:19.519,0:03:21.760
because
0:03:20.400,0:03:23.360
the wave was incident from the left
0:03:21.760,0:03:24.959
there's a broken symmetry just given
0:03:23.360,0:03:26.720
by our starting condition of sending
0:03:24.959,0:03:29.760
the particle in from the left
0:03:26.720,0:03:32.400
so in region 2 we'll have phi_transmitted
0:03:29.760,0:03:33.920
they may have a
0:03:32.400,0:03:37.280
different amplitude t
0:03:33.920,0:03:40.640
e to the i and let's call it k prime
0:03:37.280,0:03:41.840
x this doesn't mean a derivative it
0:03:40.640,0:03:43.440
is just a different label for a
0:03:41.840,0:03:45.280
different k
0:03:43.440,0:03:47.120
okay so let's just look at the
0:03:45.280,0:03:49.840
normalizations on these first
0:03:47.120,0:03:51.360
i haven't put one on here if you
0:03:49.840,0:03:53.519
recall the previous videos
0:03:51.360,0:03:54.959
these are going to be amplitudes
0:03:53.519,0:03:57.120
the modulus square of these should give
0:03:54.959,0:03:58.239
us the probability densities
0:03:57.120,0:03:59.840
and we'd like to integrate our
0:03:58.239,0:04:01.439
probability densities across all of
0:03:59.840,0:04:02.959
space to be able to get one.
0:04:01.439,0:04:04.879
Now actually the case of plane waves is
0:04:02.959,0:04:06.959
a bit unphysical and they're what
0:04:04.879,0:04:09.439
are called non-normalizable
0:04:06.959,0:04:10.319
so you can't really have a plane wave it
0:04:09.439,0:04:11.599
doesn't
0:04:10.319,0:04:13.519
it's not a valid solution to the
0:04:11.599,0:04:14.879
Schroedinger equation because those waves
0:04:13.519,0:04:16.000
are not normalizable
0:04:14.879,0:04:18.000
as you can probably imagine from the
0:04:16.000,0:04:19.759
fact that they exist across all of space
0:04:18.000,0:04:21.680
so in reality what we would form is a
0:04:19.759,0:04:22.560
wave packet which would have a finite
0:04:21.680,0:04:24.400
extent
0:04:22.560,0:04:25.759
it'd be like a plane wave for some
0:04:24.400,0:04:26.880
region of space and that thing would be
0:04:25.759,0:04:28.400
moving along
0:04:26.880,0:04:30.400
it's a more difficult case to consider
0:04:28.400,0:04:33.199
mathematically
0:04:30.400,0:04:33.600
but but it's possible to form such
0:04:33.199,0:04:34.880
things
0:04:33.600,0:04:36.479
so we're just going to consider the
0:04:34.880,0:04:37.520
easiest case of plain waves but they're
0:04:36.479,0:04:38.800
not normalizable
0:04:37.520,0:04:40.400
so our pre-factors out the front
0:04:38.800,0:04:41.840
actually don't really matter but the
0:04:40.400,0:04:45.040
relative size of them does matter so
0:04:41.840,0:04:45.040
we're going to solve for r and t
0:04:45.280,0:04:50.800
okay so we've got the the setup then
0:04:47.840,0:04:52.800
so in region one
0:04:50.800,0:04:54.000
oh and sorry we're going to
0:04:52.800,0:04:57.199
consider the case
0:04:54.000,0:04:59.440
E > V_0
0:04:57.199,0:05:00.240
hence the form of these all taking
0:04:59.440,0:05:03.280
the form of
0:05:00.240,0:05:05.440
travelling waves plane waves and in
0:05:03.280,0:05:07.520
region one
0:05:05.440,0:05:08.720
our time independent Schroedinger
0:05:07.520,0:05:12.080
equation reads
0:05:08.720,0:05:15.680
-hbar^2/2 m phi''(x)
0:05:12.080,0:05:16.639
V(x) is 0 in
0:05:15.680,0:05:20.000
that region
0:05:16.639,0:05:23.680
so this just equals E phi
0:05:20.000,0:05:27.360
if we substitute oh and sorry
0:05:23.680,0:05:30.479
and in region one phi in region one
0:05:27.360,0:05:33.600
is equal to phi_in
0:05:30.479,0:05:37.680
plus phi_reflected
0:05:33.600,0:05:42.320
and in region two phi_two is just
0:05:37.680,0:05:46.080
phi_transmitted so in region 1
0:05:42.320,0:05:49.199
we can insert phi
0:05:46.080,0:05:49.199
region 1 into here
0:05:50.080,0:05:54.000
taking 2 derivatives of this with
0:05:52.960,0:05:57.520
respect to x
0:05:54.000,0:06:00.639
brings down i k twice from both of these
0:05:57.520,0:06:01.440
and so we just find i k squared is minus
0:06:00.639,0:06:04.400
k squared
0:06:01.440,0:06:06.160
we find that hbar squared k squared
0:06:04.400,0:06:09.840
over 2m
0:06:06.160,0:06:12.639
phi 1 equals E phi
0:06:09.840,0:06:13.680
1 and we can cancel the phi 1's out and
0:06:12.639,0:06:16.800
find the energy
0:06:13.680,0:06:17.600
in region 1 is equal to
0:06:17.600,0:06:21.840
h bar squared k squared over 2m
0:06:29.120,0:06:35.600
and this is in region 1.
0:06:32.240,0:06:35.600
In region two
0:06:36.479,0:06:41.120
our Schroedinger equation reads
0:06:38.479,0:06:51.759
-hbar^2/2m phi_2'' + V_0 phi_2 = E phi_2.
0:06:52.400,0:06:58.880
Substitute our right going
0:06:55.840,0:07:02.319
travelling wave into there brings down i
0:06:58.880,0:07:06.240
k prime twice and we find that h bar
0:07:02.319,0:07:09.440
squared k prime squared over 2m
0:07:06.240,0:07:11.759
plus v_0 equals E cancelling the
0:07:09.440,0:07:14.880
phi_2s out
0:07:11.759,0:07:18.080
and so in this case we have that
0:07:14.880,0:07:19.919
k prime is equal to
0:07:18.080,0:07:22.319
multiplied by 2m take the V_0
0:07:19.919,0:07:26.080
across to m
0:07:22.319,0:07:26.080
E - V_0
0:07:26.400,0:07:32.800
square root that divided by hbar
0:07:30.639,0:07:33.919
whereas from the top equation here we
0:07:32.800,0:07:38.000
could have said that k
0:07:33.919,0:07:40.720
equals 2 m e
0:07:38.000,0:07:41.840
over h bar so k and k prime are
0:07:40.720,0:07:44.160
different because of the
0:07:41.840,0:07:45.120
the different V values the different
0:07:44.160,0:07:48.000
potential and different features oh
0:07:45.120,0:07:48.000
sorry you can't quite see that
0:07:48.160,0:07:51.680
okay so k prime and k are wave
0:07:50.639,0:07:52.800
vectors and the wave vectors are
0:07:51.680,0:07:55.840
different in the different regions
0:07:52.800,0:07:55.840
because of the different potentials
0:07:56.240,0:08:02.800
okay so
0:08:00.000,0:08:03.520
good oh yes so now what we'd like to do
0:08:02.800,0:08:05.680
is
0:08:03.520,0:08:06.960
to solve using our boundary
0:08:05.680,0:08:09.360
conditions
0:08:06.960,0:08:13.120
so let me bring this back up here so
0:08:09.360,0:08:16.080
remember our general boundary conditions
0:08:13.120,0:08:16.080
from a previous video
0:08:20.479,0:08:23.599
for the time independent Schroedinger
0:08:22.080,0:08:26.720
equation
0:08:23.599,0:08:30.800
first tells us that phi
0:08:26.720,0:08:30.800
is continuous in space
0:08:31.120,0:08:34.479
and so what this tells us in the present
0:08:33.039,0:08:37.680
case is that
0:08:34.479,0:08:40.880
phi_1 evaluated at
0:08:37.680,0:08:41.680
x=0 where the two meet equals
0:08:40.880,0:08:45.760
phi_2
0:08:41.680,0:08:45.760
at x=0
0:08:46.240,0:08:53.040
and the second condition
0:08:49.360,0:08:54.000
is that phi prime the partial derivative
0:08:53.040,0:08:58.080
of phi with respect
0:08:54.000,0:08:58.080
to position x is also continuous
0:08:59.600,0:09:03.120
and so this gives us a condition for
0:09:01.600,0:09:04.320
the derivatives we're at the point where
0:09:03.120,0:09:06.880
the wave functions meet
0:09:04.320,0:09:07.920
and we find that phi_1 prime let's
0:09:06.880,0:09:13.120
just write 0
0:09:07.920,0:09:15.600
equals phi_2 prime and 0.
0:09:13.120,0:09:17.120
okay so taking condition 1 and
0:09:15.600,0:09:21.040
substituting
0:09:17.120,0:09:23.680
the expression for our amplitudes
0:09:21.040,0:09:25.760
we have that so let me just show you
0:09:23.680,0:09:28.720
again over here
0:09:25.760,0:09:30.160
so phi one is phi_in plus phi_r so
0:09:28.720,0:09:32.000
we're going to substitute x=0
0:09:30.160,0:09:34.399
into here so we get one from this
0:09:32.000,0:09:35.360
1+r from this one and that's going to
0:09:34.399,0:09:37.839
equal t
0:09:35.360,0:09:37.839
from this one
0:09:42.560,0:09:46.160
so what we're doing is substituting the
0:09:45.200,0:09:47.920
expressions
0:09:46.160,0:09:49.200
for the wave functions into this
0:09:47.920,0:09:52.560
expression
0:09:49.200,0:09:54.959
and then this is just a typo here okay
0:09:52.560,0:09:56.080
and for two we need to take the first
0:09:54.959,0:09:59.360
derivative
0:09:56.080,0:10:00.560
of these things so we're going to bring
0:09:59.360,0:10:04.320
down an ik
0:10:00.560,0:10:07.760
from this one minus i k from this one
0:10:04.320,0:10:07.760
and then i k prime from this one
0:10:07.920,0:10:14.560
and so we have that i k
0:10:11.440,0:10:18.000
e to the i k x
0:10:14.560,0:10:21.600
minus i k r
0:10:18.000,0:10:23.120
e to the minus i k x
0:10:21.600,0:10:25.040
and that thing is going to evaluate
0:10:23.120,0:10:28.560
at x=0
0:10:25.040,0:10:32.079
is equal to i k prime
0:10:28.560,0:10:35.200
t e to the i k prime x
0:10:32.079,0:10:36.800
also evaluated at x=0 just
0:10:35.200,0:10:39.920
taking the first derivative
0:10:36.800,0:10:43.519
and so we find cancelling out the i's
0:10:39.920,0:10:46.560
we find that k times
0:10:43.519,0:10:50.079
1-r equals k
0:10:46.560,0:10:50.079
prime times t
0:10:51.120,0:10:54.640
okay or in other words we can say
0:10:53.360,0:10:59.120
that 1-r
0:10:54.640,0:11:01.200
equals k prime over k times t
0:10:59.120,0:11:02.399
so we now have enough to solve for
0:11:01.200,0:11:06.560
the amplitudes
0:11:02.399,0:11:08.480
we have two expressions
0:11:06.560,0:11:10.079
two unknowns remember k and k prime are
0:11:08.480,0:11:12.240
known because they're specified by the
0:11:10.079,0:11:15.360
energy and V_0
0:11:12.240,0:11:17.120
and so we can
0:11:15.360,0:11:21.440
take the sum of these two things to get
0:11:17.120,0:11:21.440
t and the difference to get r
0:11:24.399,0:11:29.839
okay so we're just going to take this
0:11:26.240,0:11:29.839
equation one and equation 2.
0:11:33.519,0:11:37.440
if we add them together we find that we
0:11:35.680,0:11:41.200
get
0:11:37.440,0:11:44.800
2 equals t times
0:11:41.200,0:11:47.200
1 plus k prime over k
0:11:44.800,0:11:48.640
or in other words our transmission
0:11:47.200,0:11:51.920
amplitude t
0:11:48.640,0:11:57.839
is equal to 2/(1+k'/k)
0:11:51.920,0:11:57.839
which equals
0:12:01.360,0:12:07.360
2k/(k+k')
0:12:05.920,0:12:10.399
and taking the difference of the
0:12:07.360,0:12:13.920
equations we find that
0:12:10.399,0:12:18.399
2r=t(1-k'/k)
0:12:18.959,0:12:25.839
or r equals
0:12:31.839,0:12:37.040
this and we can substitute in the value
0:12:34.480,0:12:39.839
of t that we just found
0:12:37.040,0:12:40.880
this equals the twos will cancel we
0:12:39.839,0:12:45.440
get k/(k+k')
0:12:40.880,0:12:47.279
times
0:12:45.440,0:12:49.040
let's multiply this through so we get
0:12:47.279,0:12:55.440
(k-k')/k
0:12:52.399,0:12:55.760
cancel the k's and we find that
0:12:55.440,0:12:58.880
r=(k-k')/(k+k')
0:12:58.880,0:13:05.040
so this is our
0:13:02.320,0:13:05.040
reflection
0:13:05.519,0:13:07.839
amplitude
0:13:09.839,0:13:14.639
and this this equals t
0:13:16.959,0:13:21.839
is our transmission
0:13:22.880,0:13:25.600
amplitude
0:13:28.800,0:13:31.680
so they have something to do with
0:13:30.240,0:13:33.200
probabilities but remember they're not
0:13:31.680,0:13:35.680
probabilities themselves
0:13:33.200,0:13:37.680
because amplitudes in general can be
0:13:35.680,0:13:39.519
complex numbers
0:13:37.680,0:13:41.600
they have this arbitrary global phase
0:13:39.519,0:13:44.800
associated with them
0:13:41.600,0:13:46.079
so to work out the probability of
0:13:44.800,0:13:49.440
the particle reflecting or
0:13:47.680,0:13:50.399
transmitting over the top of this
0:13:49.440,0:13:52.720
barrier
0:13:50.399,0:13:53.600
we we need to look at something
0:13:52.720,0:13:54.880
slightly different
0:13:53.600,0:13:57.600
now it turns out that the relevant
0:13:54.880,0:13:59.120
quantity to consider is our probability
0:13:57.600,0:14:00.480
current density that we saw in a
0:13:59.120,0:14:03.040
previous video
0:14:00.480,0:14:16.639
so let's head back over to the board for
0:14:03.040,0:14:16.639
a second and take a quick look at that
V2.1c Scattering from a potential step (part III)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continuing from video V2.1b) scattering from a potential step in the case that the energy of the particle is greater than that of the step; probability current densities for reflected and transmitted waves. Continued in V2.1d.
0:00:06.720,0:00:10.719
okay
0:00:07.440,0:00:11.599
so we have our phi_in our plane wave
0:00:10.719,0:00:14.160
going in
0:00:11.599,0:00:14.960
phi_r our plane wave reflecting and
0:00:14.160,0:00:16.800
phi_t
0:00:14.960,0:00:18.480
our plane wave transmitting we've solved
0:00:16.800,0:00:19.439
for the amplitudes for reflection and
0:00:18.480,0:00:21.199
transmission
0:00:19.439,0:00:22.880
to solve for the probabilities we need
0:00:21.199,0:00:23.519
to use our probability current density
0:00:22.880,0:00:26.720
we derived
0:00:23.519,0:00:30.400
in a previous video so the general form
0:00:26.720,0:00:33.520
looks like this where psi
0:00:30.400,0:00:36.160
is the time dependent wave function
0:00:33.520,0:00:37.040
in fact for all our plane waves the
0:00:36.160,0:00:40.480
time dependent
0:00:37.040,0:00:40.480
solution takes the following form
0:00:40.559,0:00:44.800
that is it's just the time
0:00:43.040,0:00:45.600
independent solution that we've been
0:00:44.800,0:00:49.760
working with
0:00:45.600,0:00:51.199
multiplied by a winding complex phase
0:00:49.760,0:00:52.800
where all the complex phases take the
0:00:51.199,0:00:54.239
same form here they wind at the same
0:00:52.800,0:00:57.120
rate given by the energy
0:00:54.239,0:00:58.719
of the particle and so we get the
0:00:57.120,0:00:59.600
expression for our probability current
0:00:58.719,0:01:02.559
density
0:00:59.600,0:01:05.680
for our time independent wave
0:01:02.559,0:01:05.680
functions of this form
0:01:06.400,0:01:09.520
that is pretty much the same form but
0:01:07.840,0:01:11.200
with phi in place of psi
0:01:09.520,0:01:13.360
so let's use this to calculate the
0:01:11.200,0:01:15.119
probabilities rather than the amplitudes
0:01:13.360,0:01:18.799
of reflection and transmission in the
0:01:15.119,0:01:18.799
two regions in another worked example
0:01:22.640,0:01:26.320
okay so let's look at the probability
0:01:25.280,0:01:29.439
current densities
0:01:26.320,0:01:29.439
in the two different regions
0:01:30.640,0:01:37.439
so from the board we had that j(x)
0:01:34.400,0:01:40.960
is equal to i h bar over 2
0:01:37.439,0:01:44.320
m phi d by dx
0:01:40.960,0:01:48.399
phi star minus phi star
0:01:44.320,0:01:51.680
d by dx phi
0:01:48.399,0:01:53.759
in region one
0:01:51.680,0:01:55.119
or rather it's actually the
0:01:53.759,0:02:00.320
reflection
0:01:55.119,0:02:04.960
part we'd like so we'd like j_reflected
0:02:00.320,0:02:07.840
is equal to as a function of x
0:02:04.960,0:02:08.720
we're going to insert phi reflected into
0:02:07.840,0:02:11.760
here
0:02:08.720,0:02:15.599
and remember phi so we've got phi_in
0:02:11.760,0:02:18.640
is equal to
0:02:15.599,0:02:22.080
e to the i k x
0:02:18.640,0:02:24.800
phi_reflected as e to the minus
0:02:22.080,0:02:27.280
i k x and that's multiplied sorry i
0:02:24.800,0:02:30.640
missed the prefactor by r
0:02:27.280,0:02:34.000
and phi transmitted is e
0:02:30.640,0:02:37.280
to the i k prime x
0:02:34.000,0:02:40.959
multiplied by t so if we stick phi_r
0:02:37.280,0:02:44.400
into here we find
0:02:40.959,0:02:47.599
i h bar over 2m out the front
0:02:44.400,0:02:48.480
let's write that in full for the
0:02:47.599,0:02:51.920
first one so we have
0:02:48.480,0:02:57.519
r e to the minus i
0:02:51.920,0:03:01.040
k x d by d x of phi star so phi star
0:02:57.519,0:03:04.239
that's d by dx
0:03:01.040,0:03:07.280
of r star
0:03:04.239,0:03:10.720
e to the plus
0:03:07.280,0:03:10.720
i k x
0:03:12.400,0:03:19.760
minus phi star is r star
0:03:16.640,0:03:23.360
e to the i k x and
0:03:19.760,0:03:27.360
d by d x of r e
0:03:23.360,0:03:27.360
to the minus i k x
0:03:29.120,0:03:36.239
and this equals i h bar over 2
0:03:32.239,0:03:38.640
m the r is a constant
0:03:36.239,0:03:39.360
so the derivative doesn't
0:03:38.640,0:03:41.040
change it
0:03:39.360,0:03:42.959
so we have: ... and that's true for both
0:03:41.040,0:03:47.040
cases so we're going to have a modulus r
0:03:42.959,0:03:50.720
squared out the front here
0:03:47.040,0:03:52.720
d by dx acting on e to the ikx brings
0:03:50.720,0:03:55.760
down an ik
0:03:52.720,0:03:57.360
so we get i k and then and it just
0:03:55.760,0:03:59.519
leaves it e to the ikx still
0:03:57.360,0:04:01.360
the e to the ikx cancels with the minus
0:03:59.519,0:04:04.400
i kx
0:04:01.360,0:04:05.599
and so that's all we get and then we
0:04:04.400,0:04:07.840
would have a minus here
0:04:05.599,0:04:09.680
e to the i k x d by d x on e to the
0:04:07.840,0:04:12.480
minus psi k x brings down a minus i
0:04:09.680,0:04:13.040
k minus and minus is plus so i actually
0:04:12.480,0:04:16.720
get a plus
0:04:13.040,0:04:20.000
ik and so we find that j_r
0:04:16.720,0:04:21.919
is equal to that's 2 i k
0:04:20.000,0:04:24.000
here 2 cancels with a 2 here we get
0:04:21.919,0:04:27.040
minus h bar
0:04:24.000,0:04:30.479
over m modulus r
0:04:27.040,0:04:33.840
squared k
0:04:30.479,0:04:36.639
if we'd have done the same for
0:04:33.840,0:04:36.639
phi_in
0:04:37.840,0:04:42.240
we would just have found that k goes to
0:04:40.400,0:04:43.759
minus k as you can see here and we lose
0:04:42.240,0:04:46.880
the modulus r squared
0:04:43.759,0:04:50.880
so we find that j_in
0:04:46.880,0:04:55.840
is equal to plus h bar over
0:04:50.880,0:04:56.960
m and we still get the k here
0:04:55.840,0:04:58.880
let's take a quick look at what we've
0:04:56.960,0:05:00.000
got here sorry let's move that over
0:04:58.880,0:05:03.039
slightly so you can see it
0:05:00.000,0:05:06.240
good so look at j_in
0:05:03.039,0:05:07.039
we have h bar k over m but remember
0:05:06.240,0:05:09.520
hbar k
0:05:07.039,0:05:10.639
from our de Broglie relation is just p
0:05:09.520,0:05:14.560
the momentum
0:05:10.639,0:05:17.280
and so this is p over m but p over m it
0:05:14.560,0:05:19.120
should just be our velocity
0:05:17.280,0:05:20.960
right classically momentum would be mass
0:05:19.120,0:05:22.560
times velocity and so
0:05:20.960,0:05:24.479
we can kind of think of this as the
0:05:22.560,0:05:25.680
velocity
0:05:24.479,0:05:26.960
or related to the velocity of the
0:05:25.680,0:05:28.800
particle remember it's a probability
0:05:26.960,0:05:31.919
current density
0:05:28.800,0:05:34.000
so it tells us how the
0:05:31.919,0:05:35.199
probability to find the particle evolves
0:05:34.000,0:05:38.320
with time but when we look we always
0:05:35.199,0:05:41.520
find the particle in one place
0:05:38.320,0:05:43.680
so the probability of reflection
0:05:41.520,0:05:47.600
let's call it R
0:05:43.680,0:05:50.160
is let's say it's defined to be
0:05:47.600,0:05:51.919
it really is this is just a sort of
0:05:50.160,0:05:54.320
physical way to motivate it it's
0:05:51.919,0:05:55.280
the probability current density for the
0:05:54.320,0:05:58.800
reflected wave
0:05:55.280,0:06:02.240
divided by the probability
0:05:58.800,0:06:05.039
current density of the in-going wave
0:06:02.240,0:06:06.400
and so in this case we get divided by
0:06:05.039,0:06:10.000
the hbar
0:06:06.400,0:06:11.840
k over m and we just get minus
0:06:10.000,0:06:13.840
r squared and sorry it should really be
0:06:11.840,0:06:14.960
the modulus of this because it's a
0:06:13.840,0:06:18.080
probability
0:06:14.960,0:06:21.840
the so we have the modulus of this
0:06:18.080,0:06:21.840
and so we get |r|^2
0:06:21.919,0:06:26.080
the minus sign of course is coming about
0:06:23.520,0:06:29.120
because j is a vector quantity
0:06:26.080,0:06:31.680
so the in-going wave has positive
0:06:29.120,0:06:32.960
velocity if you like positive current
0:06:31.680,0:06:34.400
probability current density
0:06:32.960,0:06:35.600
so the reflected wave must have a
0:06:34.400,0:06:36.240
negative one because it's a vector
0:06:35.600,0:06:37.680
quantity
0:06:36.240,0:06:39.440
so you can think of k as being negative
0:06:37.680,0:06:41.919
in this case
0:06:39.440,0:06:43.360
if we do the same thing for
0:06:41.919,0:06:48.400
phi_transmitted
0:06:43.360,0:06:50.800
we find that j_transmitted
0:06:48.400,0:06:52.319
is equal to: ... so it's rightgoing
0:06:50.800,0:06:52.880
you can do this explicitly i'm just
0:06:52.319,0:06:56.639
going to
0:06:52.880,0:07:00.080
state the result so we're going to get
0:06:56.639,0:07:05.039
a plus h bar over
0:07:00.080,0:07:08.080
m but we don't get a k we get a k prime
0:07:05.039,0:07:09.599
and we get a modulus
0:07:08.080,0:07:11.280
t squared much like we have the modulus
0:07:09.599,0:07:14.840
r squared for the other case
0:07:11.280,0:07:18.080
and so the probability of
0:07:14.840,0:07:21.199
transmission which is
0:07:18.080,0:07:25.440
defined to be j transmitted over
0:07:21.199,0:07:29.599
j incident is equal to ...
0:07:25.440,0:07:32.880
divided by h bar k over m and we get
0:07:29.599,0:07:35.599
modulus t squared k prime over
0:07:32.880,0:07:37.120
k and again sorry it's really
0:07:35.599,0:07:38.240
technically the modulus of this just in
0:07:37.120,0:07:41.840
case there's any
0:07:38.240,0:07:46.479
sign coming in there okay
0:07:41.840,0:07:48.160
so it's not just modulus t squared
0:07:46.479,0:07:49.520
the reason for the k prime of the k you
0:07:48.160,0:07:50.800
can think of it as the particle having
0:07:49.520,0:07:51.680
different velocities in the different
0:07:50.800,0:07:54.000
regions
0:07:51.680,0:07:54.960
and so you're saying more likely to find
0:07:54.000,0:07:57.680
a particle
0:07:54.960,0:07:58.080
if it's moving slowly in a region is
0:07:57.680,0:07:59.360
one
0:07:58.080,0:08:02.560
loose way to think of what's going on
0:07:59.360,0:08:05.680
here so we have the probability
0:08:02.560,0:08:08.080
for reflection and transmission
0:08:05.680,0:08:09.440
and in fact we can substitute back in
0:08:08.080,0:08:10.960
the energy E and V_0
0:08:09.440,0:08:12.479
into this expression if we want
0:08:10.960,0:08:14.479
to get it back in terms of
0:08:12.479,0:08:15.680
of the energy of the particle and
0:08:14.479,0:08:19.680
V_0
0:08:15.680,0:08:20.479
okay so let's take a look now at the
0:08:19.680,0:08:22.720
case that
0:08:20.479,0:08:27.840
E < V_0 and we'll switch
0:08:22.720,0:08:27.840
back to the board for a second
V2.1d Scattering from a potential step (part IV)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continuing from video V2.1c) scattering from a potential step in the case that the energy of the particle is less than that of the step; probability amplitudes for reflected and transmitted waves; probability current densities for reflection and transmission.
0:00:08.480,0:00:10.240
okay
0:00:08.960,0:00:16.320
so this time we're going to consider the
0:00:10.240,0:00:20.080
case E < V_0 send a
0:00:16.320,0:00:20.080
plane wave in as before from the left
0:00:22.480,0:00:26.160
and by the way when i'm drawing the wave
0:00:24.160,0:00:28.880
functions here on this plot
0:00:26.160,0:00:30.080
it's really a bit of a mix of of two
0:00:28.880,0:00:31.439
different diagrams
0:00:30.080,0:00:33.520
in the picture here this is showing the
0:00:31.439,0:00:35.440
potential so the y-axis here is the
0:00:33.520,0:00:38.320
potential as a function of position
0:00:35.440,0:00:40.480
i'm drawing the waves coming in like
0:00:38.320,0:00:41.920
this
0:00:40.480,0:00:44.079
and of course really i'm here i'm
0:00:41.920,0:00:46.079
plotting the amplitude of the
0:00:44.079,0:00:47.200
wave function in the region as a
0:00:46.079,0:00:49.280
function of position
0:00:47.200,0:00:50.800
it's a very mixed notation but
0:00:49.280,0:00:53.199
it's something we tend to do
0:00:50.800,0:00:54.719
hopefully the idea is clear that this is
0:00:53.199,0:00:55.920
a wave coming in and this just happens
0:00:54.719,0:00:57.199
to be the potential landscape it's
0:00:55.920,0:01:00.719
moving in
0:00:57.199,0:01:03.359
now this is phi_in
0:01:00.719,0:01:04.159
coming in this way as before we're going
0:01:03.359,0:01:07.199
to find
0:01:04.159,0:01:07.199
reflected coming back
0:01:11.680,0:01:15.439
but this time the because
0:01:14.479,0:01:17.759
E < V_0
0:01:15.439,0:01:19.680
past the region over here is
0:01:17.759,0:01:22.080
what's called classically forbidden
0:01:19.680,0:01:23.360
so if we just take a step back for a
0:01:22.080,0:01:24.400
second and think about what would happen
0:01:23.360,0:01:26.880
classically
0:01:24.400,0:01:27.920
the potential here could be something
0:01:26.880,0:01:29.759
like a height a height is a
0:01:27.920,0:01:33.280
gravitational potential
0:01:29.759,0:01:35.840
so we could have a height that's
0:01:33.280,0:01:36.400
zero over here and then V_0 over
0:01:35.840,0:01:38.320
here
0:01:36.400,0:01:40.720
and this would be like some kind of wall
0:01:38.320,0:01:42.640
and if we send in a classical particle
0:01:40.720,0:01:45.680
as in say throw a tennis ball
0:01:42.640,0:01:46.880
in the direction of a wall if the
0:01:45.680,0:01:50.399
tennis ball has
0:01:46.880,0:01:52.000
this energy a gravitational energy
0:01:50.399,0:01:53.280
that's larger than that of the wall as
0:01:52.000,0:01:56.159
in it's going over the wall
0:01:53.280,0:01:56.880
then sure it can go over here but if
0:01:56.159,0:02:00.079
we send in
0:01:56.880,0:02:01.920
a tennis ball whose gravitational
0:02:00.079,0:02:03.520
potential energy is less than that of
0:02:01.920,0:02:04.000
the top of the wall it should bounce
0:02:03.520,0:02:05.360
back
0:02:04.000,0:02:07.040
of course this is then classically
0:02:05.360,0:02:08.319
forbidden tennis ball can't go into the
0:02:07.040,0:02:10.720
wall
0:02:08.319,0:02:12.720
in fact we'll see that in the
0:02:10.720,0:02:14.480
Schroedinger equation it's allowed
0:02:12.720,0:02:17.040
the form of the solutions are now our
0:02:14.480,0:02:19.280
evanescent type solutions
0:02:17.040,0:02:20.080
exponentially growing or decreasing and
0:02:19.280,0:02:21.840
in this case
0:02:20.080,0:02:24.319
only the decreasing solution can be
0:02:21.840,0:02:24.319
relevant
0:02:24.720,0:02:27.840
again not worrying too much about the
0:02:26.480,0:02:30.560
fact that i'm mixing
0:02:27.840,0:02:31.599
my notations with the potential and the
0:02:30.560,0:02:32.959
wave function
0:02:31.599,0:02:36.640
so we'll have an exponentially
0:02:32.959,0:02:38.800
decreasing amplitude over here
0:02:36.640,0:02:40.239
rather that's right an exponentially
0:02:38.800,0:02:43.360
decreasing
0:02:40.239,0:02:44.480
wave function because otherwise
0:02:43.360,0:02:46.400
the wave function would have to blow up
0:02:44.480,0:02:48.840
at infinity if we allow the positive
0:02:46.400,0:02:50.239
solution this barrier goes up to
0:02:48.840,0:02:53.360
infinity
0:02:50.239,0:02:53.680
okay so let's take a look at solving
0:02:53.360,0:02:55.599
this
0:02:53.680,0:02:58.879
explicitly and looking at the
0:02:55.599,0:02:58.879
classically forbidden region
0:03:02.800,0:03:05.760
okay so we'll go a little bit faster
0:03:04.480,0:03:07.760
than we did in the last one but the
0:03:05.760,0:03:11.120
working is very much the same
0:03:07.760,0:03:13.840
so in region one
0:03:11.120,0:03:14.800
we'll have phi_1 which is equal to
0:03:13.840,0:03:20.640
phi_in + phi_reflected
0:03:18.239,0:03:21.360
and this can be written as e to the i k x
0:03:21.360,0:03:27.120
plus r e to the minus i k x just as
0:03:24.560,0:03:29.360
before nothing's changing in region one
0:03:27.120,0:03:30.400
substitute into the time independent
0:03:29.360,0:03:33.440
Schroedinger equation
0:03:30.400,0:03:34.159
and we'll find that e equals h bar
0:03:33.440,0:03:37.120
squared k
0:03:34.159,0:03:39.120
squared over 2m just like before in
0:03:37.120,0:03:42.159
region 2 however
0:03:39.120,0:03:42.720
we'll have phi in region 2 is equal to
0:03:42.159,0:03:45.040
phi
0:03:42.720,0:03:46.000
transmitted remember t for transmitted
0:03:45.040,0:03:48.879
not for
0:03:46.000,0:03:49.599
time or anything like that and this is
0:03:48.879,0:03:52.799
equal to
0:03:49.599,0:03:56.319
t e to the minus
0:03:52.799,0:03:58.239
kappa x; kappa used
0:03:56.319,0:03:59.599
instead of k just to indicate that it's
0:03:58.239,0:04:01.680
really a different quantity to a wave
0:03:59.599,0:04:03.360
vector
0:04:01.680,0:04:05.280
substitutes into the time independent
0:04:03.360,0:04:08.640
Schroedinger equation in region 2
0:04:05.280,0:04:09.680
and we find that e equals minus h bar
0:04:08.640,0:04:12.959
squared
0:04:09.680,0:04:16.239
kappa squared over 2m plus
0:04:12.959,0:04:18.479
V_0 so
0:04:16.239,0:04:19.840
the minus here coming from the fact
0:04:18.479,0:04:21.440
that there was a minus ready in the
0:04:19.840,0:04:23.360
time independent Schroedinger equation
0:04:21.440,0:04:24.720
and we don't cancel it with a an i
0:04:23.360,0:04:28.000
squared as we do
0:04:24.720,0:04:30.880
in the case of plane waves okay
0:04:28.000,0:04:33.280
so boundary conditions as before in the
0:04:30.880,0:04:36.240
first boundary condition is that the
0:04:33.280,0:04:37.120
wave function is continuous so phi in
0:04:36.240,0:04:39.440
region one
0:04:37.120,0:04:40.479
at x=0 is equal to phi in
0:04:39.440,0:04:43.759
region two
0:04:40.479,0:04:47.440
at x=0 which tells us that
0:04:43.759,0:04:50.560
1+r=t just like before
0:04:47.440,0:04:50.960
boundary condition 2 the first oh dear
0:04:50.560,0:04:53.440
sorry
0:04:50.960,0:04:54.840
the first derivative of the wave
0:04:53.440,0:04:57.280
function must also
0:04:54.840,0:04:59.840
match the first derivative is also
0:04:57.280,0:04:59.840
continuous
0:04:59.919,0:05:03.600
and in this case what we find this gives
0:05:02.560,0:05:07.280
us is that
0:05:03.600,0:05:07.520
i k (1 - r) that's what we got before
0:05:07.280,0:05:10.560
on
0:05:07.520,0:05:14.160
in region 1 is now equal to minus
0:05:10.560,0:05:16.080
kappa t in region 2.
0:05:14.160,0:05:18.240
so slightly different form in fact
0:05:16.080,0:05:21.440
rewriting we have
0:05:18.240,0:05:25.360
1-r equals multiply through by minus
0:05:21.440,0:05:30.560
i we get i kappa over k
0:05:25.360,0:05:33.600
t just as before we have two equations
0:05:30.560,0:05:34.960
two unknowns r and t the amplitudes
0:05:33.600,0:05:35.600
for reflection and transmission around
0:05:34.960,0:05:38.080
the kappa
0:05:35.600,0:05:39.600
and k are both already known from
0:05:38.080,0:05:42.000
kappa from this equation k from this
0:05:39.600,0:05:42.000
equation
0:05:42.080,0:05:46.400
and using these two equations
0:05:45.039,0:05:49.680
we can derive
0:05:46.400,0:05:53.280
the probability of reflection which is k
0:05:49.680,0:05:56.240
minus i kappa over k
0:05:53.280,0:05:57.199
plus i kappa and the probability of
0:05:56.240,0:06:00.800
transmission
0:05:57.199,0:06:03.759
is 2 k/(k+i kappa)
0:06:00.800,0:06:04.800
in fact if you compare to
0:06:03.759,0:06:06.400
the results we got
0:06:04.800,0:06:08.560
for the case that energy was greater
0:06:06.400,0:06:10.000
than V_0 you'll see that
0:06:08.560,0:06:12.160
because all we've really done is we've
0:06:10.000,0:06:16.160
changed what was e to the
0:06:12.160,0:06:18.720
i k prime x to e to the minus kappa x
0:06:16.160,0:06:19.440
we've just changed i k prime to minus
0:06:18.720,0:06:20.639
kappa
0:06:19.440,0:06:23.600
and if you make those substitutions
0:06:20.639,0:06:26.240
you'll find you get the same results
0:06:23.600,0:06:27.840
all right so let's flip the page around
0:06:26.240,0:06:29.039
and look at the probabilities so those
0:06:27.840,0:06:30.000
are the amplitudes
0:06:29.039,0:06:32.479
actually let's just comment on this
0:06:30.000,0:06:35.840
quickly so we have the amplitude
0:06:32.479,0:06:37.120
for the particle to exist in region two
0:06:35.840,0:06:38.720
is non-zero
0:06:37.120,0:06:41.120
so remember region two is classically
0:06:38.720,0:06:42.800
forbidden but there's an
0:06:41.120,0:06:44.240
amplitude to find the particle there
0:06:42.800,0:06:45.680
what that means is that if we perform a
0:06:44.240,0:06:47.039
measurement we could find the particle
0:06:45.680,0:06:50.000
in region two
0:06:47.039,0:06:51.440
how does this tally
0:06:50.000,0:06:53.680
with the fact that there isn't enough
0:06:51.440,0:06:54.960
energy for the particle to be there
0:06:53.680,0:06:57.120
the answer is that our measurement
0:06:54.960,0:06:58.000
device will provide the energy if
0:06:57.120,0:06:59.759
it's to observe
0:06:58.000,0:07:01.680
the particle and really what we're doing
0:06:59.759,0:07:03.520
is changing the boundary conditions on
0:07:01.680,0:07:04.720
the problem we're changing the potential
0:07:03.520,0:07:06.000
when we make the measurement we'll
0:07:04.720,0:07:07.280
see more of that in the quantum
0:07:06.000,0:07:10.639
tunnelling
0:07:07.280,0:07:10.639
problem in the next video
0:07:11.759,0:07:15.759
nevertheless we let's take a look at
0:07:14.400,0:07:18.000
the probability
0:07:15.759,0:07:19.599
for the probability current densities in
0:07:18.000,0:07:24.000
the different regions
0:07:19.599,0:07:24.000
so let me just write this down like this
0:07:24.960,0:07:31.440
there we go partial subscript
0:07:28.160,0:07:32.080
x again means d phi* in this case by
0:07:31.440,0:07:35.039
dx
0:07:32.080,0:07:35.039
holding time constant
0:07:35.199,0:07:38.800
and so what we get is that phi
0:07:37.680,0:07:42.319
for phi_in
0:07:38.800,0:07:44.639
which equals e to the i k x
0:07:42.319,0:07:45.440
substitutes into here and just as before
0:07:44.639,0:07:48.560
we find
0:07:45.440,0:07:51.840
j_in equals h bar k over
0:07:48.560,0:07:54.960
m phi
0:07:51.840,0:07:57.840
transmitted equals the
0:07:54.960,0:07:58.479
transmission amplitude e to the minus
0:07:57.840,0:08:01.280
kappa
0:07:58.479,0:08:04.400
x substitute that into here let me just
0:08:01.280,0:08:04.400
write it out like this again
0:08:05.680,0:08:10.319
so we need this we substitute phi and
0:08:09.039,0:08:13.840
phi* into here
0:08:10.319,0:08:13.840
evaluating it we find this
0:08:15.759,0:08:19.120
so we bring down a minus kappa in both
0:08:18.639,0:08:20.720
cases
0:08:19.120,0:08:22.479
but now the minus sign between the two
0:08:20.720,0:08:24.800
terms remember it's this term
0:08:22.479,0:08:26.240
minus the complex conjugate but the two
0:08:24.800,0:08:28.639
terms are the same because phi
0:08:26.240,0:08:30.479
is now real and so actually we find that
0:08:28.639,0:08:33.279
these perfectly cancel out
0:08:30.479,0:08:34.240
we find that j_transmitted=0
0:08:33.279,0:08:37.760
and so therefore
0:08:34.240,0:08:41.519
the probability of transmission
0:08:37.760,0:08:44.640
which is equal to |j_t/j_in|
0:08:41.519,0:08:46.160
is equal to zero
0:08:44.640,0:08:48.640
this is what we call the
0:08:46.160,0:08:50.240
probability flux
0:08:48.640,0:08:51.920
the probability that the probability
0:08:50.240,0:08:53.839
current is going to
0:08:53.839,0:08:56.959
a traveling wave in the classically
0:08:56.000,0:08:58.160
forbidden region
0:08:56.959,0:09:00.080
so even though there's an amplitude to
0:08:58.160,0:09:01.760
find the particle over there
0:09:00.080,0:09:03.360
the probability current
0:09:01.760,0:09:04.880
density is zero so there's no
0:09:03.360,0:09:06.800
current propagating through that
0:09:04.880,0:09:09.200
region if we look at
0:09:06.800,0:09:11.360
the probability for reflection which is
0:09:09.200,0:09:14.160
|j_reflected / j_incident|
0:09:11.360,0:09:15.279
in fact the expression is the
0:09:14.160,0:09:17.440
same as we found
0:09:15.279,0:09:19.519
in the last video still
0:09:17.440,0:09:22.399
|r|^2
0:09:19.519,0:09:23.760
and if we substitute it in we find
0:09:22.399,0:09:28.640
for r we had
0:09:23.760,0:09:32.160
(k- i kappa)/(k+ i kappa)
0:09:28.640,0:09:34.480
and if you do the modulus square of this
0:09:32.160,0:09:37.519
you'll find out that it equals one
0:09:34.480,0:09:39.120
which must have been the case because
0:09:37.519,0:09:40.560
the probability of transmission plus
0:09:39.120,0:09:41.920
the probability of reflection is always
0:09:40.560,0:09:44.240
equal to one
0:09:41.920,0:09:47.519
from conservation of probability okay
0:09:44.240,0:09:47.519
thank you for your time
V2.2 Quantum tunnelling
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
quantum tunnelling (aka barrier penetration), scattering over a finite-width potential barrier, and resonant transmission.
0:00:00.560,0:00:04.480
hello in this video we're going to be
0:00:02.720,0:00:07.600
looking at quantum tunnelling
0:00:04.480,0:00:09.760
here's the potential of the problem and
0:00:07.600,0:00:11.280
let's draw it
0:00:09.760,0:00:13.280
so it's very much like the potential
0:00:11.280,0:00:14.000
step we considered in the previous two
0:00:13.280,0:00:15.360
videos
0:00:14.000,0:00:17.359
except for rather than going off to
0:00:15.360,0:00:20.800
infinity at positive x
0:00:17.359,0:00:24.560
it comes back down again to zero
0:00:20.800,0:00:26.720
in potential so
0:00:24.560,0:00:28.320
let the form of the solutions again
0:00:26.720,0:00:30.080
depend on which region we're in
0:00:28.320,0:00:32.000
let's call these regions one two and
0:00:30.080,0:00:34.399
three
0:00:32.000,0:00:35.600
and the solutions in one two and
0:00:34.399,0:00:37.200
three will depend on whether
0:00:35.600,0:00:38.879
the energy of the particle is greater
0:00:37.200,0:00:39.600
than the barrier height or less than the
0:00:38.879,0:00:41.520
barrier height
0:00:39.600,0:00:43.280
just like before when it's greater we'll
0:00:41.520,0:00:45.520
have plane waves in all
0:00:43.280,0:00:47.039
three regions when it's less
0:00:45.520,0:00:49.680
than the barrier height we'll have plane
0:00:47.039,0:00:52.960
waves in regions one and three
0:00:49.680,0:00:56.320
but evanescent waves in region two
0:00:52.960,0:00:56.320
so we can draw them something like this
0:00:56.719,0:01:00.480
I won't draw the plane waves because we
0:00:58.160,0:01:02.879
know what those look like.
0:01:00.480,0:01:03.760
For the energy less than the barrier
0:01:02.879,0:01:06.960
height
0:01:03.760,0:01:10.840
we'll have a plane wave coming in
0:01:06.960,0:01:12.080
and then we'll have either exponentially
0:01:10.840,0:01:15.040
growing
0:01:12.080,0:01:16.320
or exponentially decreasing solutions in
0:01:15.040,0:01:19.680
the barrier
0:01:16.320,0:01:21.360
and then a plane wave over here
0:01:19.680,0:01:23.439
and again I'd like to reiterate that
0:01:21.360,0:01:25.600
when I draw these pictures of the
0:01:23.439,0:01:27.200
wave functions of course they're on
0:01:25.600,0:01:30.240
different axes.
0:01:27.200,0:01:32.400
They don't have the same y-axis here
0:01:30.240,0:01:33.439
it's just a convenient schematic
0:01:32.400,0:01:34.720
notation
0:01:33.439,0:01:36.400
the other thing that's a bit
0:01:34.720,0:01:37.040
misleading about it is that of course
0:01:36.400,0:01:38.560
these are the
0:01:37.040,0:01:40.799
time independent solutions but I've just
0:01:38.560,0:01:43.040
taken them at a particular time
0:01:40.799,0:01:44.640
as time evolves the phase of these
0:01:43.040,0:01:45.360
waves changes so you can think of moving
0:01:44.640,0:01:46.720
along
0:01:45.360,0:01:48.320
and so this point we're moving up and
0:01:46.720,0:01:48.720
down so the fact that I draw them matching
0:01:48.320,0:01:50.799
here
0:01:48.720,0:01:52.159
is really just a convention and in fact
0:01:50.799,0:01:54.079
you can't even draw them matched
0:01:52.159,0:01:55.759
at the other end note that we're going
0:01:54.079,0:01:57.600
to need both the exponentially
0:01:55.759,0:02:00.159
increasing and decreasing
0:01:57.600,0:02:02.719
evanescent solutions in this case in
0:02:00.159,0:02:04.880
order to match the boundary conditions
0:02:02.719,0:02:05.920
okay so let's take a look at the form of
0:02:04.880,0:02:08.800
the waves in each
0:02:05.920,0:02:11.440
region let's clear the board good in
0:02:08.800,0:02:14.000
region one we have the following form
0:02:11.440,0:02:14.800
that is just like with the potential
0:02:14.000,0:02:16.800
step problem
0:02:14.800,0:02:18.879
we're going to send in a plane wave from
0:02:16.800,0:02:21.520
the left we'll set the
0:02:18.879,0:02:23.280
amplitude to one by convention and we'll
0:02:21.520,0:02:26.640
get a reflected wave travelling
0:02:23.280,0:02:28.879
back in the left direction
0:02:26.640,0:02:30.000
reflected back from the barrier. In
0:02:28.879,0:02:31.840
region three
0:02:30.000,0:02:34.560
that is on this side of the barrier
0:02:31.840,0:02:37.200
we'll have the following form
0:02:34.560,0:02:38.160
so a transmitted wave heading over to
0:02:37.200,0:02:40.959
the right again
0:02:38.160,0:02:42.640
but this time note that k is the same
0:02:40.959,0:02:44.239
k as appears in phi_1
0:02:42.640,0:02:45.840
this is because the potential is the
0:02:44.239,0:02:47.440
same in both regions
0:02:45.840,0:02:50.239
the potential is set to zero in both
0:02:47.440,0:02:52.959
regions so in particular the
0:02:50.239,0:02:55.120
energy eigenvalues in these regions are
0:02:52.959,0:02:56.959
as follows
0:02:55.120,0:02:58.159
that is the energy is h bar squared k
0:02:56.959,0:03:00.480
squared over 2m
0:02:58.159,0:03:01.920
for both regions one and three so it
0:03:00.480,0:03:03.440
really is the same k
0:03:01.920,0:03:05.360
in region two it depends on whether the
0:03:03.440,0:03:06.800
energy is greater than or less than
0:03:05.360,0:03:08.080
V_0 for the case that energy is
0:03:06.800,0:03:10.000
greater than V_0 we have plane
0:03:08.080,0:03:12.239
waves
0:03:10.000,0:03:13.200
where k prime is different now
0:03:12.239,0:03:14.879
because it solves
0:03:13.200,0:03:16.840
the energy eigenvalues of the time
0:03:14.879,0:03:18.480
independent Schroedinger equation are as
0:03:16.840,0:03:20.480
follows
0:03:18.480,0:03:23.120
so because of this additional V_0,
0:03:20.480,0:03:24.480
k prime does not equal k
0:03:23.120,0:03:26.799
and when E < V_0 we
0:03:24.480,0:03:29.920
have evanescent waves
0:03:26.799,0:03:32.239
of this form and the energy eigenvalues
0:03:29.920,0:03:34.799
look like this
0:03:32.239,0:03:35.680
that is we have a minus sign in front of
0:03:34.799,0:03:38.959
the
0:03:35.680,0:03:42.080
h bar squared kappa squared over 2m
0:03:38.959,0:03:43.200
and this then ensures that for
0:03:42.080,0:03:46.879
real kappa
0:03:43.200,0:03:47.920
E - V_0 is negative or
0:03:46.879,0:03:49.599
V_0 - E
0:03:47.920,0:03:51.519
is positive which is true when the
0:03:49.599,0:03:54.400
energy is less than V_0
0:03:51.519,0:03:55.120
and up here we have real k prime
0:03:54.400,0:03:58.400
giving
0:03:55.120,0:04:01.680
E - V_0 is positive so
0:03:58.400,0:04:03.599
notice that if we can solve the case for...
0:04:01.680,0:04:04.799
so quantum tunnelling occurs for when E
0:04:03.599,0:04:06.000
is less than V_0
0:04:04.799,0:04:08.080
we have to get through a classically
0:04:06.000,0:04:08.720
forbidden region when E>V_0
0:04:08.720,0:04:12.640
we're simply scattering over a potential
0:04:10.560,0:04:14.560
barrier of finite width
0:04:12.640,0:04:16.639
it's a tiny bit easier to solve
0:04:14.560,0:04:18.560
they're pretty much the same
0:04:16.639,0:04:19.680
in terms of difficulty I know that we
0:04:18.560,0:04:21.919
can solve
0:04:19.680,0:04:23.360
the case of E > V_0
0:04:21.919,0:04:24.960
and then get the solutions for
0:04:23.360,0:04:28.560
E < V_0 for free
0:04:24.960,0:04:31.040
by making the following observation
0:04:28.560,0:04:32.800
we can just substitute ik'
0:04:31.040,0:04:34.400
in place of kappa
0:04:32.800,0:04:36.160
and that will switch the form of the
0:04:34.400,0:04:38.400
solution from here to here
0:04:36.160,0:04:39.759
and it will also make this change here
0:04:38.400,0:04:40.479
so we can solve one of these problems
0:04:39.759,0:04:42.240
and get the other
0:04:40.479,0:04:44.000
solutions for free so let's deal with
0:04:42.240,0:04:46.880
the plane waves predominantly
0:04:44.000,0:04:48.800
this a tiny bit easier so let's do our
0:04:46.880,0:04:51.919
usual thing we have to just match the
0:04:48.800,0:04:53.199
wave functions at the boundaries using
0:04:51.919,0:04:56.400
the boundary conditions
0:04:53.199,0:05:00.560
we now have four unknowns
0:04:56.400,0:05:02.479
r, t, a, and b, but we also have
0:05:00.560,0:05:04.639
four matching conditions so let's
0:05:02.479,0:05:06.800
write those down
0:05:04.639,0:05:10.400
so first let's just reproduce the
0:05:06.800,0:05:13.520
picture a little bit smaller down here
0:05:10.400,0:05:16.080
so first we have that the wave function
0:05:13.520,0:05:18.880
must be continuous in space
0:05:16.080,0:05:20.639
so at x = 0 we have to match
0:05:18.880,0:05:22.160
wave functions one and two
0:05:20.639,0:05:24.320
and they must be equal to each other
0:05:22.160,0:05:26.800
substituting this into
0:05:24.320,0:05:30.080
the forms of the wave
0:05:26.800,0:05:32.720
functions we find the following result
0:05:30.080,0:05:33.360
1 + r = a + b
0:05:32.720,0:05:34.639
next we have to match
0:05:33.360,0:05:37.440
make sure that the derivatives are
0:05:34.639,0:05:39.039
continuous substituting we find this
0:05:37.440,0:05:41.360
result
0:05:39.039,0:05:42.160
and we have the same two conditions
0:05:41.360,0:05:45.440
to apply
0:05:42.160,0:05:45.440
at x = L
0:05:46.080,0:05:49.440
continuous wave function continuous
0:05:47.680,0:05:52.560
derivative and these gives the
0:05:49.440,0:05:54.880
following two conditions
0:05:52.560,0:05:57.520
okay so it's not particularly intuitive
0:05:54.880,0:05:59.840
we have four equations and four unknowns
0:05:57.520,0:06:01.520
and we can solve for them we'll do
0:05:59.840,0:06:04.560
so in the problem sets
0:06:01.520,0:06:06.160
well I'll just take the solution here
0:06:04.560,0:06:08.240
in particular we're interested in say
0:06:06.160,0:06:11.199
the transmission amplitude t
0:06:08.240,0:06:13.039
to get into this region and in
0:06:11.199,0:06:15.600
particular we're really interested in
0:06:13.039,0:06:17.520
the probability of transmission
0:06:15.600,0:06:19.360
which remember is given by the
0:06:17.520,0:06:21.919
ratio of the
0:06:19.360,0:06:23.919
probability current density in the
0:06:21.919,0:06:26.080
transmitted region to
0:06:23.919,0:06:29.039
the probability current density ingoing
0:06:27.280,0:06:30.800
and if we evaluate that for the case
0:06:29.039,0:06:31.680
that E > V_0 to plane
0:06:30.800,0:06:36.080
waves everywhere
0:06:31.680,0:06:37.919
we find the following result so
0:06:36.080,0:06:39.280
the probability of transmission the
0:06:37.919,0:06:41.440
ratio of
0:06:39.280,0:06:43.039
the probability current density in
0:06:41.440,0:06:45.280
region 3 to that
0:06:43.039,0:06:46.479
ingoing in this case it is just
0:06:45.280,0:06:48.800
|t|^2
0:06:46.479,0:06:50.160
the the velocities of the particles you
0:06:48.800,0:06:51.199
can think of as the same in regions one
0:06:50.160,0:06:53.360
and three because the
0:06:51.199,0:06:54.880
potentials are the same and we get
0:06:53.360,0:06:57.360
this expression here
0:06:54.880,0:06:58.160
now probably the most interesting
0:06:57.360,0:07:00.960
thing about it
0:06:58.160,0:07:02.240
is that we have this
0:07:00.960,0:07:06.160
sin^2(k'L) term
0:07:02.240,0:07:10.400
meaning that whenever
0:07:06.160,0:07:12.160
k'L = n pi for integer n
0:07:10.400,0:07:13.759
we have what's called
0:07:12.160,0:07:16.960
'resonant transmission'
0:07:13.759,0:07:17.680
meaning when this is fulfilled this
0:07:16.960,0:07:20.560
is zero
0:07:17.680,0:07:22.720
this whole thing is zero and we have
0:07:20.560,0:07:25.919
probability of transmission equals one
0:07:22.720,0:07:27.680
so we can tune E,
0:07:25.919,0:07:29.919
V_0, or L to get this condition to
0:07:27.680,0:07:30.479
be fulfilled and it's rather
0:07:29.919,0:07:32.080
interesting
0:07:30.479,0:07:34.160
perhaps the more philosophically
0:07:32.080,0:07:38.080
profound case is when E < V_0
0:07:34.160,0:07:38.800
so a similar-looking expression
0:07:38.080,0:07:41.280
to this
0:07:38.800,0:07:42.000
except for we now have a sinh
0:07:41.280,0:07:43.680
instead of a sine
0:07:42.000,0:07:45.599
so we lose our resonant transmission
0:07:43.680,0:07:48.720
condition sorry let me just
0:07:45.599,0:07:50.000
make a note of that name so
0:07:48.720,0:07:51.199
with E > V_0 we can have
0:07:50.000,0:07:53.280
resonant transmission
0:07:51.199,0:07:54.960
when e is less than V_0 we can't
0:07:53.280,0:07:56.400
but it's somewhat miraculous that we can
0:07:54.960,0:07:57.520
even have transmission at all
0:07:56.400,0:07:59.759
given that we have to pass through a
0:07:57.520,0:08:02.639
barrier which is classically forbidden
0:07:59.759,0:08:04.319
so quantum objects can quantum
0:08:02.639,0:08:05.039
mechanically tunnel through barriers to
0:08:04.319,0:08:07.840
which they couldn't
0:08:05.039,0:08:09.919
normally pass classically so this is the
0:08:07.840,0:08:13.120
basis of an experimental technique
0:08:09.919,0:08:16.960
called scanning tunnelling microscopy
0:08:13.120,0:08:19.759
you bring down a measurement tip
0:08:16.960,0:08:21.680
close to a sample and you can by
0:08:19.759,0:08:23.199
applying a bias voltage you can tunnel
0:08:21.680,0:08:23.759
electrons either from the tip into the
0:08:23.199,0:08:26.000
sample
0:08:23.759,0:08:27.120
or from the sample into the tip and
0:08:26.000,0:08:28.560
what tends to be done there are
0:08:27.120,0:08:29.599
different methods of operating this but
0:08:28.560,0:08:32.719
they tend to run it in
0:08:29.599,0:08:33.919
the constant current mode in which you
0:08:33.919,0:08:37.120
vary the height of your tip in order to
0:08:35.599,0:08:39.039
maintain the same current
0:08:37.120,0:08:41.120
coming through as you move the tip along
0:08:39.039,0:08:42.959
so then you record the height of the tip
0:08:41.120,0:08:45.120
at different places and you can use this
0:08:42.959,0:08:45.839
to map out the effective height of the
0:08:45.120,0:08:48.000
surface
0:08:45.839,0:08:49.680
down to the atomic scale so there's a
0:08:48.000,0:08:52.800
let me get a picture up for you here
0:08:49.680,0:08:54.000
so here's a picture of an STM image as
0:08:52.800,0:08:55.360
it's called
0:08:54.000,0:08:56.880
where you're really seeing individual
0:08:55.360,0:08:59.040
atoms so it's an incredibly powerful
0:08:56.880,0:09:01.200
technique
0:08:59.040,0:09:03.200
this also resolves a bit of a paradox we
0:09:01.200,0:09:05.040
had in the previous two videos where we
0:09:03.200,0:09:07.600
looked at scattering from
0:09:05.040,0:09:08.480
an infinitely long step and we saw that
0:09:07.600,0:09:10.720
there was no
0:09:08.480,0:09:12.160
probability for transmission in the
0:09:10.720,0:09:13.040
sense of probability current getting
0:09:12.160,0:09:14.560
through
0:09:13.040,0:09:16.560
but there was nevertheless an amplitude
0:09:14.560,0:09:17.839
for transmission and I said at the time
0:09:16.560,0:09:18.320
that what happens is if you were to
0:09:17.839,0:09:19.920
measure
0:09:18.320,0:09:21.920
a particle there you could find one
0:09:19.920,0:09:23.680
there but you'd be providing the energy
0:09:21.920,0:09:25.760
using your measurement device
0:09:23.680,0:09:27.519
so now we see what happens really
0:09:25.760,0:09:28.640
you're changing the form of the solution
0:09:27.519,0:09:29.680
originally we had the step that went off
0:09:28.640,0:09:31.120
to infinity
0:09:29.680,0:09:32.640
when you bring a measurement device in
0:09:31.120,0:09:33.440
you're actually changing the potential
0:09:32.640,0:09:35.600
landscape
0:09:33.440,0:09:37.680
you're making it possible for the
0:09:35.600,0:09:38.720
particle to exist inside the detector
0:09:37.680,0:09:40.399
and so you're bringing the step back
0:09:38.720,0:09:42.240
down again
0:09:40.399,0:09:43.920
so you're really turning the infinite
0:09:42.240,0:09:46.240
step problem into
0:09:43.920,0:09:46.959
the finite length barrier problem and
0:09:46.240,0:09:49.279
that's how
0:09:46.959,0:09:50.480
you're able to measure the particle in
0:09:49.279,0:09:53.760
that region
0:09:50.480,0:09:55.279
so I'm going to show you it so
0:09:53.760,0:09:56.880
this seems like an incredibly quantum
0:09:55.279,0:09:59.200
phenomenon and it is really
0:09:56.880,0:10:00.720
but there is a precedent for it in
0:09:59.200,0:10:03.200
terms of evanescent waves
0:10:00.720,0:10:04.079
in light and I'll show you that in a
0:10:03.200,0:10:07.200
separate video
0:10:04.079,0:10:07.200
thank you
V2.3 Evanescent waves demo
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
a demonstration of coupling to evanescent light waves in order to transform an amplitude for transmission into a fully fledged probability current density. As Maxwell's equations are compatible with quantum mechanics this can either be described as a classical wave effect or the quantum tunnelling of photons through a classically-forbidden region. The demo is followed by a brief discussion of why quantum tunnelling is so magical.
Music: Angelo Badalamenti - Audrey's Dance (Twin Peaks OST)
0:00:00.719,0:00:04.400
Hello welcome to my kitchen where I'm
0:00:03.120,0:00:06.240
going to show a quick demonstration
0:00:04.400,0:00:06.720
about quantum tunnelling and evanescent
0:00:06.240,0:00:09.679
waves
0:00:06.720,0:00:10.080
using light so what I have here is a
0:00:09.679,0:00:13.440
nice
0:00:10.080,0:00:16.080
block of perspex
0:00:13.440,0:00:17.680
and let me just turn the light off here
0:00:16.080,0:00:19.520
wait for the camera to adjust
0:00:17.680,0:00:21.119
so I'm going to shine a laser into the
0:00:19.520,0:00:23.519
block I found that
0:00:21.119,0:00:24.560
red is probably the most effective for
0:00:23.519,0:00:27.199
showing this
0:00:24.560,0:00:29.519
so if I shine the laser at a nice steep
0:00:27.199,0:00:31.519
angle like this
0:00:29.519,0:00:33.600
I get a spot coming through on the back
0:00:31.519,0:00:34.880
you can see over here here's a nice spot
0:00:33.600,0:00:36.000
and you can even see the beam going
0:00:34.880,0:00:37.360
through and you can see some of it
0:00:36.000,0:00:38.800
reflecting so there's some
0:00:37.360,0:00:40.559
probability for reflection some
0:00:38.800,0:00:43.520
probability for transmission
0:00:40.559,0:00:44.239
if I increase the angle of incidence
0:00:43.520,0:00:46.559
like this
0:00:44.239,0:00:47.920
eventually there we go I get total
0:00:46.559,0:00:50.399
internal reflection
0:00:47.920,0:00:52.160
so there's now no beam coming through at
0:00:50.399,0:00:52.719
all the probability of transmission is
0:00:52.160,0:00:54.160
zero
0:00:52.719,0:00:56.079
there's a bit of ambient light coming
0:00:54.160,0:00:57.440
from around the sides here
0:00:56.079,0:00:59.359
but just to show you here's what happens
0:00:57.440,0:01:02.000
when a full beam goes through
0:00:59.359,0:01:03.760
can you see that it's much bigger and
0:01:02.000,0:01:05.199
here's where it's disappeared again
0:01:03.760,0:01:08.560
it's totally internal
0:01:05.199,0:01:10.560
reflected there okay
0:01:08.560,0:01:12.159
so when we're getting total internal
0:01:10.560,0:01:14.320
reflection the probability
0:01:12.159,0:01:16.400
for transmission the probability current
0:01:14.320,0:01:19.280
density on the other side
0:01:16.400,0:01:20.240
is zero but there's nevertheless an
0:01:19.280,0:01:23.600
amplitude
0:01:20.240,0:01:26.159
to detect a photon out the back there
0:01:23.600,0:01:26.799
so what we could do in order to show
0:01:26.159,0:01:29.520
that
0:01:26.799,0:01:32.079
is remember in the quantum problem if
0:01:29.520,0:01:35.520
we have the infinitely long
0:01:32.079,0:01:36.799
barrier then we won't get any
0:01:35.520,0:01:37.840
probability current density in that
0:01:36.799,0:01:39.439
region
0:01:37.840,0:01:41.360
but if we can make the barrier finite
0:01:39.439,0:01:42.240
length by coupling to some measurement
0:01:41.360,0:01:43.920
device
0:01:42.240,0:01:45.840
we can actually get a probability
0:01:43.920,0:01:48.000
current in that other region
0:01:45.840,0:01:49.759
so what I'd like to show you is I'd like
0:01:48.000,0:01:52.159
to take this prism
0:01:49.759,0:01:53.040
right here and I'd like to place it
0:01:52.159,0:01:56.320
behind
0:01:53.040,0:01:59.920
the block over here and
0:01:56.320,0:02:02.240
leave a little air gap and have the
0:01:59.920,0:02:03.280
prism coupled to the evanescent wave out
0:02:02.240,0:02:05.680
the back of the
0:02:03.280,0:02:07.520
perspex block and and take some of the
0:02:05.680,0:02:09.679
reflected power away and divert it
0:02:07.520,0:02:12.160
and make a transmitted wave now that's
0:02:09.679,0:02:14.480
not going to happen
0:02:12.160,0:02:15.200
so there's the beam going
0:02:14.480,0:02:17.520
through
0:02:15.200,0:02:19.599
here's total internal reflection and
0:02:17.520,0:02:21.440
here's me placing the prism here
0:02:19.599,0:02:23.440
and it doesn't steal any of the
0:02:21.440,0:02:25.280
reflected power
0:02:23.440,0:02:26.560
it's not because quantum mechanics is
0:02:25.280,0:02:30.239
wrong it's just because
0:02:26.560,0:02:31.440
the wavelength of this laser is 650
0:02:30.239,0:02:35.040
nanometers
0:02:31.440,0:02:37.280
so in order to
0:02:35.040,0:02:38.879
couple to that exponentially dying
0:02:37.280,0:02:42.800
evanescent wave I'd need to get
0:02:38.879,0:02:45.760
within a few wavelengths of the light
0:02:42.800,0:02:48.080
and 650 nanometers is somewhere between
0:02:45.760,0:02:51.120
a 20th and a 200th of the
0:02:48.080,0:02:52.800
width of a hair so I'm not going to
0:02:51.120,0:02:53.360
realistically get this prism close
0:02:52.800,0:02:55.920
enough
0:02:53.360,0:02:58.480
to the block in order to couple to it
0:02:55.920,0:03:00.400
to take any significant power away
0:02:58.480,0:03:02.319
I can do a bit of a cheat though which
0:03:00.400,0:03:03.680
is that I need to find a material
0:03:02.319,0:03:05.440
or some kind of surface that I can get
0:03:03.680,0:03:07.200
close enough to the back of this
0:03:05.440,0:03:08.720
perspex block that I can couple the
0:03:07.200,0:03:12.159
evanescent wave inside
0:03:08.720,0:03:12.959
off the back and the trick is that
0:03:12.159,0:03:15.519
I can just
0:03:12.959,0:03:18.400
pour some water in there because of
0:03:15.519,0:03:19.440
course water is going to be
0:03:18.400,0:03:21.920
let's set up the total internal
0:03:19.440,0:03:23.440
reflection the water is going to get
0:03:21.920,0:03:26.799
close enough to the back there right
0:03:23.440,0:03:26.799
now it's a bit of a cheat
0:03:27.680,0:03:34.879
whoopsy daisy let's get that back there
0:03:31.519,0:03:34.879
hopefully when it settles down
0:03:36.560,0:03:40.959
I think we need a little bit more water
0:03:38.080,0:03:40.959
just to bring the height up
0:03:49.840,0:03:52.400
there we go so now you can see on the
0:03:50.959,0:03:53.599
back wall we're getting that beam
0:03:52.400,0:03:55.280
through can you see that
0:03:53.599,0:03:57.120
there we go so the reason it's a little
0:03:55.280,0:03:58.799
bit of a cheat is just that
0:03:57.120,0:04:00.400
you know what I'm really doing is
0:03:58.799,0:04:02.239
just changing the refractive index of
0:04:00.400,0:04:03.360
the material out the back of the perspex
0:04:02.239,0:04:05.280
block
0:04:03.360,0:04:06.640
but that kind of explains why this
0:04:05.280,0:04:09.840
had to work, right?
0:04:06.640,0:04:11.360
Because you know that if I put a
0:04:09.840,0:04:13.040
higher refractive index material out the
0:04:11.360,0:04:14.799
back like water I will change the
0:04:13.040,0:04:15.439
critical angle and get a beam to go
0:04:14.799,0:04:18.400
through
0:04:15.439,0:04:20.079
but how if all the individual photons
0:04:18.400,0:04:21.519
were reflecting before how would they
0:04:20.079,0:04:23.600
know to go through if you change the
0:04:21.519,0:04:25.360
stuff behind the perspex block
0:04:23.600,0:04:28.000
and the answer is that there's actually
0:04:25.360,0:04:31.199
a probability sorry an amplitude
0:04:28.000,0:04:32.639
for transmission but since I'm using
0:04:31.199,0:04:35.840
water here
0:04:32.639,0:04:38.000
let me just turn this light on and so
0:04:35.840,0:04:39.120
sorry and the reason light is so good
0:04:38.000,0:04:42.240
for showing this is
0:04:39.120,0:04:43.919
as we said before that light
0:04:42.240,0:04:45.600
is it can be thought of either as
0:04:43.919,0:04:46.400
quantum or classical. Maxwell's
0:04:45.600,0:04:48.479
equations
0:04:46.400,0:04:49.919
are compatible with quantum mechanics
0:04:48.479,0:04:53.600
so you can either think of this
0:04:49.919,0:04:55.120
as a beam of light
0:04:53.600,0:04:56.560
in a classical manner or you can think
0:04:55.120,0:04:57.919
of that beam of light as being made up
0:04:56.560,0:05:01.199
of individual quanta of
0:04:57.919,0:05:02.479
energy called photons and and those
0:05:01.199,0:05:04.160
descriptions will be compatible with
0:05:02.479,0:05:06.800
each other so you can think of this
0:05:04.160,0:05:07.280
as an evanescent wave for classical
0:05:06.800,0:05:08.720
light
0:05:07.280,0:05:10.560
or you can think of it as quantum
0:05:08.720,0:05:12.880
tunnelling of the photons through
0:05:10.560,0:05:14.000
the classically forbidden region as it
0:05:12.880,0:05:17.280
were
0:05:14.000,0:05:19.199
okay so since I've got the cup of water
0:05:17.280,0:05:21.600
here let me just adjust the focus on the
0:05:19.199,0:05:21.600
camera
0:05:22.080,0:05:26.720
there we go so there's actually a
0:05:25.199,0:05:29.120
an even better demonstration you can do
0:05:26.720,0:05:30.560
with water you can see down here that we
0:05:29.120,0:05:32.479
have total internal reflection in the
0:05:30.560,0:05:32.960
cup you can't see the tip of my finger
0:05:32.479,0:05:34.720
right
0:05:32.960,0:05:36.000
here you can see it and down here you
0:05:34.720,0:05:38.080
can't because
0:05:36.000,0:05:40.960
there's total internal reflection if I
0:05:38.080,0:05:43.039
just get it a tiny bit damp
0:05:40.960,0:05:44.320
and I place my finger on the back
0:05:43.039,0:05:46.720
here
0:05:44.320,0:05:47.840
you see my fingerprint come about that's
0:05:46.720,0:05:50.000
quite clear isn't it
0:05:47.840,0:05:51.840
so what's happening is that the
0:05:50.000,0:05:52.800
ridges of my fingerprint are getting
0:05:51.840,0:05:55.919
close enough
0:05:52.800,0:05:56.960
to the water and the
0:05:55.919,0:05:58.560
plastic
0:05:56.960,0:05:59.680
that they can couple to that evanescent
0:05:58.560,0:06:00.960
wave so I should be getting total
0:05:59.680,0:06:04.479
internal reflection
0:06:00.960,0:06:06.000
a probability of reflection 1
0:06:04.479,0:06:07.360
but by placing something close enough to
0:06:06.000,0:06:08.639
the back I can actually couple to the
0:06:07.360,0:06:11.360
evanescent wave and I can
0:06:08.639,0:06:12.639
steal some of that power away so as some
0:06:11.360,0:06:14.800
of the reflective power and it turns
0:06:12.639,0:06:16.720
into transmitted and so then I get a
0:06:14.800,0:06:18.160
propagating wave out the back
0:06:16.720,0:06:20.080
but you can see how sensitive it is
0:06:18.160,0:06:22.479
because the
0:06:20.080,0:06:24.720
troughs of my fingerprint are too far
0:06:22.479,0:06:27.520
away to have any significant coupling
0:06:24.720,0:06:29.120
hence you see dark where sorry see
0:06:27.520,0:06:30.240
light where the troughs are because I'm
0:06:29.120,0:06:31.440
not coupling and you've got total
0:06:30.240,0:06:33.360
internal reflection
0:06:31.440,0:06:34.560
and you see dark where the the peaks of
0:06:33.360,0:06:36.560
my fingerprint are
0:06:34.560,0:06:38.240
because that is coupling to the
0:06:36.560,0:06:38.560
evanescent wave and directing the power
0:06:38.240,0:06:41.120
out
0:06:38.560,0:06:44.000
so I'm getting quantum tunnelling into
0:06:41.120,0:06:46.400
my finger of these individual photons
0:06:44.000,0:06:48.319
so you can think of it with light either
0:06:46.400,0:06:50.800
as a classical or a quantum effect
0:06:48.319,0:06:52.800
it doesn't make the quantum problem any
0:06:50.800,0:06:55.440
less magical it just means that
0:06:52.800,0:06:56.000
classical waves are more magical
0:06:55.440,0:06:59.120
let's
0:06:56.000,0:07:03.840
take a look at some of that
0:06:59.120,0:07:03.840
magic in a different room
0:07:06.800,0:07:10.319
In terms of a classical particle however
0:07:08.639,0:07:13.680
it's really quite weird
0:07:10.319,0:07:17.120
imagine we take a box like this
0:07:13.680,0:07:19.120
which contains two halves separated by a
0:07:17.120,0:07:21.360
finite potential barrier
0:07:19.120,0:07:22.800
we take a classical particle such as
0:07:21.360,0:07:26.000
this marble
0:07:22.800,0:07:26.000
place it into one of the halves
0:07:26.400,0:07:31.840
and no matter how much shaking up we
0:07:27.840,0:07:31.840
give it
0:07:34.960,0:07:45.840
we'll always find it in that same half
0:07:47.680,0:07:50.960
imagine now it's a quantum particle
0:07:49.120,0:07:52.879
however and when we
0:07:50.960,0:07:54.319
give it the same shaking up half the
0:07:52.879,0:07:58.650
time we might expect to find it
0:07:54.319,0:08:01.730
in the other half
0:08:04.319,0:08:08.400
depending on whether the outside of the
0:08:06.160,0:08:13.840
box counts as a finite barrier or not
0:08:08.400,0:08:13.840
I might even expect to find
0:08:18.400,0:08:28.400
it's not in the box at all.
0:08:26.319,0:08:28.400
Thank you.
V3.1 The infinite potential well
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
energy eigenfunctions and eigenvalues of the infinite potential well (aka particle in a 1D box).
0:00:00.399,0:00:04.799
hi in this video we're going to take a
0:00:02.480,0:00:07.040
look at the infinite potential well
0:00:04.799,0:00:08.880
also sometimes known as particle in a
0:00:07.040,0:00:12.320
box
0:00:08.880,0:00:15.360
the potential is as follows so
0:00:12.320,0:00:17.760
0 within the region [0,L] along the
0:00:15.360,0:00:19.039
1d line or infinity outside of that
0:00:17.760,0:00:22.080
region
0:00:19.039,0:00:24.080
and let's draw it so
0:00:22.080,0:00:25.840
when we say the potential is infinity
0:00:24.080,0:00:27.840
here and here this really just means
0:00:25.840,0:00:30.560
that the particle can't exist there
0:00:27.840,0:00:32.000
so our boundary conditions take a
0:00:30.560,0:00:34.480
slightly different form this time we
0:00:32.000,0:00:36.480
have that
0:00:34.480,0:00:37.760
the
0:00:36.480,0:00:39.680
wave function at
0:00:37.760,0:00:40.879
position x=0 and at
0:00:39.680,0:00:42.960
position x=L
0:00:40.879,0:00:44.559
must be equal to zero so we're no longer
0:00:42.960,0:00:46.000
looking for the wave function to be
0:00:44.559,0:00:48.800
continuous between the regions
0:00:46.000,0:00:50.640
we use the fact that the wave function
0:00:48.800,0:00:52.079
must vanish in these two regions here
0:00:50.640,0:00:54.160
and so must vanish on the boundaries of
0:00:52.079,0:00:56.160
those regions
0:00:54.160,0:00:57.760
so within the well itself where we want
0:00:56.160,0:00:59.520
to solve for the particle
0:00:57.760,0:01:01.520
we have the potential equal zero it's
0:00:59.520,0:01:03.199
still a constant so we're going to have
0:01:01.520,0:01:05.280
our
0:01:03.199,0:01:06.799
plane wave type solutions but in this
0:01:05.280,0:01:07.760
case we're going to be considering
0:01:06.799,0:01:09.680
standing waves
0:01:07.760,0:01:13.439
rather than travelling waves so the
0:01:09.680,0:01:16.320
solutions look like this
0:01:13.439,0:01:16.720
causing the sign I mean you can apply
0:01:16.320,0:01:18.080
these
0:01:16.720,0:01:19.840
boundary conditions I've labeled them
0:01:18.080,0:01:21.119
both 1 because that's they're kind of
0:01:19.840,0:01:22.640
taking the place of
0:01:21.119,0:01:24.960
what used to be condition 1 let's call
0:01:22.640,0:01:30.799
them condition 1
0:01:24.960,0:01:30.799
and condition 2.
0:01:30.880,0:01:34.000
so applying those boundary conditions to
0:01:33.280,0:01:37.840
this state
0:01:34.000,0:01:40.479
boundary condition 1 tells us
0:01:37.840,0:01:40.880
that phi of zero equals zero so if we
0:01:40.479,0:01:42.720
stick
0:01:40.880,0:01:44.159
x=0 into here this one
0:01:42.720,0:01:48.240
disappears anyway
0:01:44.159,0:01:48.240
and we find that 0=A
0:01:52.560,0:01:55.840
and using condition 2
0:01:57.360,0:02:00.560
we only have B sin(k x) left we
0:02:00.159,0:02:03.040
stick
0:02:00.560,0:02:03.600
x=L into it find it must equal
0:02:03.040,0:02:07.200
zero
0:02:03.600,0:02:10.640
and so we need sin(k L)=0
0:02:07.200,0:02:14.239
and so we find that
0:02:10.640,0:02:18.319
k L = n pi
0:02:14.239,0:02:21.599
where n is any integer
0:02:18.319,0:02:24.879
so let's label those k's by that integer
0:02:21.599,0:02:26.239
n and our solutions for the
0:02:24.879,0:02:30.080
eigenenergies
0:02:26.239,0:02:32.400
are as follows h bar squared k squared
0:02:30.080,0:02:34.640
over 2m but where k is now labeled by n
0:02:32.400,0:02:35.440
and we can substitute this expression
0:02:34.640,0:02:39.280
in here for
0:02:35.440,0:02:41.920
kn so this equals
0:02:39.280,0:02:46.400
hbar^2/2m (n pi /L)^2
0:02:43.200,0:02:48.000
so and let's label the energies by
0:02:46.400,0:02:50.160
that integer n as well
0:02:48.000,0:02:52.480
so we have an infinite tower of
0:02:50.160,0:02:54.480
different energy eigenstates which solve
0:02:52.480,0:02:57.200
the Schroedinger equation in the well
0:02:54.480,0:02:58.879
given by this expression here
0:02:57.200,0:03:00.560
for each there's a corresponding
0:02:58.879,0:03:01.760
eigenfunction so these are the energy
0:03:00.560,0:03:03.519
eigenvalues that solve the time
0:03:01.760,0:03:05.599
independent Schroedinger equation
0:03:03.519,0:03:06.640
and our energy eigenfunctions we've
0:03:05.599,0:03:09.920
solved for
0:03:06.640,0:03:13.360
here let's just rewrite this equation
0:03:09.920,0:03:13.360
taking these two things into account
0:03:13.680,0:03:17.519
so this is our time independent solution
0:03:16.400,0:03:18.000
the solution to the time independent
0:03:17.519,0:03:20.720
Schroedinger
0:03:18.000,0:03:22.560
equation as always we can add back in
0:03:20.720,0:03:23.920
our time dependence very easily for the
0:03:22.560,0:03:26.480
energy eigenvalues
0:03:23.920,0:03:27.360
and in fact if we do that we remember
0:03:26.480,0:03:29.200
that write the time
0:03:27.360,0:03:31.040
dependent wave function as psi rather
0:03:29.200,0:03:33.680
than phi and we can update this as
0:03:31.040,0:03:35.599
follows
0:03:33.680,0:03:37.440
so when we add the time dependence back
0:03:35.599,0:03:40.080
in we can also label our
0:03:37.440,0:03:41.280
energy eigenstates by n our energy
0:03:40.080,0:03:44.000
eigenfunctions
0:03:41.280,0:03:46.319
we have the time independent form
0:03:44.000,0:03:49.440
multiplied by our winding phase factor
0:03:46.319,0:03:49.920
e^(- i E_n t / hbar) where
0:03:49.440,0:03:53.120
E_n
0:03:49.920,0:03:56.400
are the energy eigenvalues okay so let's
0:03:53.120,0:03:56.799
plot those solutions on the next
0:03:56.400,0:04:01.840
board
0:03:56.799,0:04:01.840
so we'll just move this up to the corner
0:04:02.080,0:04:07.840
so for n equals one we have this form
0:04:05.840,0:04:11.040
with plus the modulus of psi one which
0:04:07.840,0:04:14.720
is time independent
0:04:11.040,0:04:15.599
similarly psi two with a node now in the
0:04:14.720,0:04:19.840
center
0:04:15.599,0:04:21.519
psi 3 and so on and each additional
0:04:19.840,0:04:23.520
as n increases we increase the number of
0:04:21.519,0:04:26.800
nodes in the well if we plot
0:04:23.520,0:04:30.400
the potential again over there
0:04:26.800,0:04:33.840
it is conventional to plot these
0:04:30.400,0:04:34.479
energy eigenfunctions on this plot now
0:04:33.840,0:04:37.919
of course
0:04:34.479,0:04:41.280
that's slightly incorrect because the
0:04:37.919,0:04:42.000
y axis here is the potential whereas the
0:04:41.280,0:04:44.639
y-axis over
0:04:42.000,0:04:45.440
here is the modulus of the wave function
0:04:44.639,0:04:48.000
even worse
0:04:45.440,0:04:49.199
we sometimes like to write what we
0:04:48.000,0:04:52.320
should write down as
0:04:49.199,0:04:53.680
psi without the modulus sign on it which
0:04:52.320,0:04:56.960
would in this case come
0:04:53.680,0:04:58.240
through down like this
0:04:56.960,0:04:59.840
but of course that's not really accurate
0:04:58.240,0:05:01.600
because psi without the modulus is a
0:04:59.840,0:05:03.680
complex number and we have this
0:05:01.600,0:05:04.880
complex winding so what you might prefer
0:05:03.680,0:05:06.560
to think
0:05:04.880,0:05:09.039
if we take the example of psi_2 for
0:05:06.560,0:05:10.400
example and rewrite it
0:05:09.039,0:05:11.680
we might write to like to write
0:05:10.400,0:05:13.280
something like this where we've drawn a
0:05:11.680,0:05:16.000
sine
0:05:13.280,0:05:16.560
but as that complex phase winds this can
0:05:16.000,0:05:19.759
of course
0:05:16.560,0:05:21.759
switch through to this solution
0:05:19.759,0:05:24.880
into negative sine but in general
0:05:21.759,0:05:28.240
actually what we really have here is
0:05:24.880,0:05:30.240
the real part of psi plotted here
0:05:28.240,0:05:32.560
and we have another axis coming out of
0:05:30.240,0:05:36.000
the board at 90 degrees which is
0:05:32.560,0:05:37.919
the imaginary part of
0:05:36.000,0:05:39.840
psi_2 in this case and really what
0:05:37.919,0:05:42.479
happens is that this
0:05:39.840,0:05:44.479
original wave function winds around
0:05:42.479,0:05:47.520
the axis in the complex plane
0:05:44.479,0:05:50.560
as it evolves in time okay all that
0:05:47.520,0:05:54.800
taken into account we can plot
0:05:50.560,0:05:59.919
the psi_1 over there
0:05:54.800,0:06:02.400
psi_2 where this is kind of a snapshot
0:05:59.919,0:06:04.800
of the complex wave function where it
0:06:02.400,0:06:07.280
happens to be purely real and
0:06:04.800,0:06:08.319
the positive value if this distance
0:06:07.280,0:06:10.560
here is one
0:06:08.319,0:06:11.919
so this the height of this would be
0:06:10.560,0:06:14.160
E_1
0:06:11.919,0:06:15.600
and height of this one would be E_2 again
0:06:14.160,0:06:17.919
this is all really conventional
0:06:15.600,0:06:19.039
the energies can be separated
0:06:17.919,0:06:20.800
according to potential they have the
0:06:19.039,0:06:23.600
same units
0:06:20.800,0:06:25.280
but the the form of the wave function
0:06:23.600,0:06:28.400
doesn't really fit on this plot
0:06:25.280,0:06:31.199
and finally psi_3
0:06:28.400,0:06:32.479
we're again drawing a snapshot of this
0:06:31.199,0:06:34.240
where happens to be real
0:06:32.479,0:06:36.319
now I'm partly showing this just because
0:06:34.240,0:06:38.479
this is so common to show these things
0:06:36.319,0:06:40.240
that you'll see it in textbooks and on
0:06:38.479,0:06:42.319
wikipedia for example
0:06:40.240,0:06:43.759
but it's slightly sloppy notation but
0:06:42.319,0:06:45.039
hopefully one that's nevertheless quite
0:06:43.759,0:06:47.039
intuitive
0:06:45.039,0:06:48.479
okay so in the next video we're going
0:06:47.039,0:06:48.960
to take a look at some properties of
0:06:48.479,0:06:51.199
these
0:06:48.960,0:06:52.080
different energy eigenfunctions that
0:06:51.199,0:06:54.400
solve the
0:06:52.080,0:06:56.240
problem in particular they're a
0:06:54.400,0:06:57.599
form of what's called a bound state
0:06:56.240,0:07:00.160
so all these states are bound into the
0:06:57.599,0:07:02.000
well there's an infinite number of them
0:07:00.160,0:07:04.479
when we call it a well even though it's
0:07:02.000,0:07:05.280
above zero in terms of potential you can
0:07:04.479,0:07:06.639
think of it as
0:07:05.280,0:07:08.319
well we can just shift down the infinite
0:07:06.639,0:07:09.680
parts down to zero and this is an
0:07:08.319,0:07:10.240
infinitely deep well with an infinite
0:07:09.680,0:07:12.319
number
0:07:10.240,0:07:14.720
of these energy eigenfunctions in it
0:07:12.319,0:07:17.840
each with their own energy eigenvalue
0:07:14.720,0:07:17.840
okay thanks your time
V3.2 Normalisation
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
finding the normalisations of the infinite potential well energy eigenstates.
0:00:00.480,0:00:03.199
hello in this video we're going to take
0:00:02.560,0:00:05.359
a look at
0:00:03.199,0:00:08.480
normalisation of the wave function
0:00:05.359,0:00:12.240
sometimes spelled with the z here
0:00:08.480,0:00:14.160
so we know from previous videos that the
0:00:12.240,0:00:16.400
modulus square of the wave function
0:00:14.160,0:00:17.520
gives us the probability density at a
0:00:16.400,0:00:19.199
point
0:00:17.520,0:00:20.640
and integrated across all of space we
0:00:19.199,0:00:22.240
expect this to give us one
0:00:20.640,0:00:24.160
because although we're not sure where
0:00:22.240,0:00:25.199
the particle is we know that it must
0:00:24.160,0:00:26.640
exist somewhere
0:00:25.199,0:00:30.080
mathematically we can write this down in
0:00:26.640,0:00:31.760
the following way
0:00:30.080,0:00:34.239
which we saw previously leads to the
0:00:31.760,0:00:37.280
conservation of global probability
0:00:34.239,0:00:39.920
let's take a look at a worked example
0:00:37.280,0:00:40.320
treating again the infinite potential
0:00:39.920,0:00:44.160
well
0:00:40.320,0:00:47.520
from video V3.1 where we'll use this
0:00:44.160,0:00:50.719
to establish the pre-factor
0:00:47.520,0:00:50.719
on the wave function
0:00:55.360,0:01:02.559
okay so we've previously seen
0:00:59.199,0:01:04.400
the infinite potential well and
0:01:02.559,0:01:07.040
just to resketch it here
0:01:04.400,0:01:08.720
we have a potential that looks like this
0:01:07.040,0:01:10.960
let's say this is x=0 and this
0:01:08.720,0:01:13.840
is x=L
0:01:10.960,0:01:13.840
this is the potential
0:01:14.479,0:01:17.600
and it's infinity in these two regions
0:01:16.880,0:01:21.439
and zero in
0:01:17.600,0:01:24.840
in the middle here we've seen that the
0:01:21.439,0:01:28.240
energy eigenstates take the form
0:01:24.840,0:01:40.000
phi_n(x)=B sin(n pi x/L)
0:01:36.240,0:01:42.159
which comes about
0:01:40.000,0:01:43.439
from requiring that the wave function
0:01:42.159,0:01:45.360
vanish at
0:01:43.439,0:01:47.920
the points where the potential goes
0:01:45.360,0:01:50.000
to infinity
0:01:47.920,0:01:53.280
so the question is what's this
0:01:50.000,0:01:55.040
coefficient here this pre-factor
0:01:53.280,0:01:57.360
it wasn't determined by the boundary
0:01:55.040,0:02:00.560
conditions but actually we can always
0:01:57.360,0:02:01.119
use the normalisation the fact
0:02:00.560,0:02:03.759
that
0:02:01.119,0:02:05.119
the integral of the probability
0:02:03.759,0:02:08.239
density across all of space
0:02:05.119,0:02:09.840
equals one to solve for the
0:02:08.239,0:02:12.640
prefactor out the front
0:02:09.840,0:02:13.680
so the condition we have is that one has
0:02:12.640,0:02:15.520
to equal
0:02:13.680,0:02:16.879
the integral of the probability density
0:02:15.520,0:02:20.560
which is given by
0:02:16.879,0:02:22.080
|phi_n(x)|^2
0:02:20.560,0:02:24.319
we integrate from minus infinity to
0:02:22.080,0:02:27.280
infinity but the wave function is zero
0:02:24.319,0:02:28.480
everywhere except for [0,L] so we
0:02:27.280,0:02:32.319
can just integrate
0:02:28.480,0:02:35.200
over [0,L]
0:02:32.319,0:02:37.200
and this is enough to solve for B up to
0:02:35.200,0:02:40.400
a global phase
0:02:37.200,0:02:43.200
so to do it we need to use
0:02:40.400,0:02:43.920
well so let's substitute this in first so
0:02:43.200,0:02:47.519
we have
0:02:43.920,0:02:48.959
the integral from 0 to L of |B|^2
0:02:47.519,0:02:50.720
because remember it can be
0:02:48.959,0:02:54.000
complex in general
0:02:50.720,0:02:57.760
sin^2( n pi x/L)
0:02:59.840,0:03:03.680
now to do this integral of
0:03:02.400,0:03:06.640
sin^2
0:03:03.680,0:03:07.360
and in general integrals for problems
0:03:06.640,0:03:09.680
to do with the
0:03:07.360,0:03:11.599
infinite well we're going to use a
0:03:09.680,0:03:14.720
couple of relations so we can use
0:03:11.599,0:03:19.920
cos^2(theta) + sin^2(theta) = 1
0:03:18.400,0:03:23.680
and the other one that tends to be useful is
0:03:19.920,0:03:28.480
cos^2(theta) - sin^2(theta) = cos(2theta)
0:03:29.120,0:03:32.640
so we have a sine squared so we want to
0:03:30.720,0:03:33.599
take this one minus this one and divide
0:03:32.640,0:03:35.599
by two
0:03:33.599,0:03:37.519
so we have that one equals we can bring
0:03:35.599,0:03:40.159
the B squared out of the integral
0:03:37.519,0:03:42.720
and actually we're going to have a
0:03:40.159,0:03:42.720
half as well
0:03:43.760,0:03:47.680
so i'm just going to take this equation
0:03:44.959,0:03:50.080
the top one and subtract this equation
0:03:47.680,0:03:51.280
and then divide by 2 to get sine squared
0:03:50.080,0:03:56.159
so we're going to have
0:03:51.280,0:03:56.159
1 - cos(2 theta)
0:03:59.200,0:04:02.239
so we have one
0:04:01.519,0:04:05.920
equals
0:04:02.239,0:04:08.959
|B|^2/2
0:04:05.920,0:04:08.959
this one just gives us L
0:04:09.040,0:04:14.080
and then we have minus integral
0:04:12.000,0:04:17.040
|B|^2 / 2
0:04:14.080,0:04:17.759
integrate this and we have sine sorry
0:04:17.040,0:04:22.880
i've put
0:04:17.759,0:04:22.880
theta here so sorry 2 theta
0:04:25.280,0:04:30.240
is equal to 2 n pi x / L
0:04:31.360,0:04:37.680
so this is going to integrate to
0:04:34.639,0:04:42.479
sin(2 theta) which is sin(2 n pi x/L)
0:04:37.680,0:04:46.880
and we need to divide by
0:04:42.479,0:04:49.440
2 and pi and multiply by L
0:04:46.880,0:04:51.040
and stick in the limit sorry to L but we
0:04:49.440,0:04:53.520
see that
0:04:51.040,0:04:54.160
when we put the limit 0 in
0:04:53.520,0:04:57.120
there that's
0:04:54.160,0:04:57.759
0 because sin(0)=0 we put
0:04:57.120,0:05:01.280
the L in
0:04:57.759,0:05:04.720
we get 2 n pi and the L to cancel
0:05:01.280,0:05:07.360
but 2 n pi for integer n
0:05:04.720,0:05:08.240
sin( 2 n pi ) is always zero so
0:05:07.360,0:05:10.320
actually this thing
0:05:08.240,0:05:11.919
is always equal to zero and so we've
0:05:10.320,0:05:15.199
solved and found
0:05:11.919,0:05:18.880
that |B|^2
0:05:15.199,0:05:23.759
is equal to 2/L or
0:05:18.880,0:05:26.880
|B| = sqrt(2/L)
0:05:23.759,0:05:30.880
and therefore we found our normalization
0:05:26.880,0:05:32.479
our energy eigenstates for
0:05:30.880,0:05:35.520
the infinite potential well
0:05:32.479,0:05:38.880
are given by
0:05:35.520,0:05:41.919
sqrt(2/L) sin( n pi x/L)
0:05:38.880,0:05:43.840
and now the probability to find
0:05:41.919,0:05:46.400
any particle described by this wave
0:05:43.840,0:05:50.720
function across all of space
0:05:46.400,0:05:50.720
is equal to one which is what we like
0:05:51.120,0:05:56.960
okay so thank you for your time
V3.3 Stationary states
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
showing that energy eigenstates have time-independent probability densities.
0:00:02.639,0:00:05.040
hello
0:00:03.600,0:00:06.960
in this rather brief video we're going
0:00:05.040,0:00:08.639
to take a look at stationary states
0:00:06.960,0:00:11.280
when we write down a solution to our
0:00:08.639,0:00:14.160
time dependent Schroedinger equation
0:00:11.280,0:00:15.200
psi(x,t) we're always
0:00:14.160,0:00:17.680
bearing in mind
0:00:15.200,0:00:19.920
that we're writing this in a separable
0:00:17.680,0:00:24.000
form
0:00:19.920,0:00:27.039
phi(x) multiplied by T(t)
0:00:24.000,0:00:29.359
we saw in the first lecture
0:00:27.039,0:00:31.279
the first set of videos that we can
0:00:29.359,0:00:32.000
always solve for T(t) and find that it
0:00:31.279,0:00:34.399
gives us
0:00:32.000,0:00:35.120
a complex phase winding since we're
0:00:34.399,0:00:36.880
looking now
0:00:35.120,0:00:38.719
at bound states we can label our
0:00:36.880,0:00:40.399
eigenstates by an integer
0:00:38.719,0:00:41.760
n labeling the states there's an
0:00:40.399,0:00:44.000
infinite number of them in the
0:00:41.760,0:00:45.280
infinite potential well and then our
0:00:44.000,0:00:47.520
general solution takes the following
0:00:45.280,0:00:47.520
form
0:00:47.760,0:00:51.680
so we can label our wave function psi by
0:00:50.480,0:00:54.640
the integer n
0:00:51.680,0:00:55.440
phi is labeled by the same n and our
0:00:54.640,0:01:00.160
phase winding
0:00:55.440,0:01:03.199
is given by the energy eigenvalue E_n
0:01:00.160,0:01:05.600
so if we look at the probability density
0:01:03.199,0:01:05.600
for this
0:01:05.920,0:01:11.360
given by |psi|^2 we
0:01:09.040,0:01:14.479
find that for these energy
0:01:11.360,0:01:16.799
states we use the complex phase and we
0:01:14.479,0:01:19.759
simply find that
0:01:16.799,0:01:20.960
we have the modulus square of the time
0:01:19.759,0:01:23.040
independent part
0:01:20.960,0:01:24.080
so this is why we say that energy
0:01:23.040,0:01:25.920
eigenstates
0:01:24.080,0:01:28.000
are what are called stationary states
0:01:25.920,0:01:31.040
the probability density for them is
0:01:28.000,0:01:32.880
constant in time
0:01:31.040,0:01:34.159
and when we say energy eigenstates this
0:01:32.880,0:01:36.320
is synonymous with
0:01:34.159,0:01:38.159
energy eigenfunctions this statement
0:01:36.320,0:01:39.200
holds completely generally it's not just
0:01:38.159,0:01:41.680
for bound states
0:01:39.200,0:01:43.119
energy eigenstates are always stationary
0:01:41.680,0:01:46.320
states
0:01:43.119,0:01:47.360
as you can derive from the substitution
0:01:46.320,0:01:49.759
of this ansatz
0:01:47.360,0:01:53.200
into your Schroedinger equation okay
0:01:49.759,0:01:53.200
thank you for your time
V3.4 Orthonormality of eigenstates
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
showing that the energy eigenstates of the infinite potential well are orthogonal and normalised (orthonormal).
0:00:00.399,0:00:04.480
hello in this video we're going to take
0:00:02.240,0:00:07.759
a look at the orthonormality
0:00:04.480,0:00:11.200
of different energy eigenstates so
0:00:07.759,0:00:14.920
we have from the previous video that all
0:00:11.200,0:00:16.720
physical states in quantum mechanics are
0:00:14.920,0:00:20.320
normalized
0:00:16.720,0:00:23.439
but in addition for any two different
0:00:20.320,0:00:24.480
energy eigenstates of the infinite
0:00:23.439,0:00:26.560
potential well
0:00:24.480,0:00:29.199
are also also orthogonal to each other
0:00:26.560,0:00:32.399
by which we mean the following
0:00:29.199,0:00:34.480
the integral of phi_n* . phi_m
0:00:32.399,0:00:36.160
over all of space in this case over
0:00:34.480,0:00:40.559
the range of the well
0:00:36.160,0:00:40.559
is equal to zero if n doesn't equal m
0:00:40.719,0:00:44.000
together these conditions tell us
0:00:42.559,0:00:46.640
that the states are
0:00:44.000,0:00:48.239
both normalized and orthogonal and we
0:00:46.640,0:00:51.520
abbreviate this to 'orthonormal'
0:00:48.239,0:00:51.520
so the statement is as follows
0:00:51.760,0:00:55.600
so the integral of phi_n* phi_m over
0:00:54.640,0:00:57.600
all of space
0:00:55.600,0:00:59.840
is equal to the kronecker delta which is
0:00:57.600,0:01:02.399
defined to be 1 if n=m
0:00:59.840,0:01:04.159
and 0 if n doesn't equal n. This is an
0:01:02.399,0:01:08.080
incredibly useful relation
0:01:04.159,0:01:11.119
which we'll put to much use in the coming
0:01:08.080,0:01:12.960
videos it doesn't just hold for energy
0:01:11.119,0:01:13.840
eigenstates of the infinite potential
0:01:12.960,0:01:16.320
well it holds
0:01:13.840,0:01:17.520
for a much broader class of problems and
0:01:16.320,0:01:19.759
we'll see much more of that
0:01:17.520,0:01:21.360
when we come to look at matrix mechanics
0:01:19.759,0:01:23.200
later on in the course
0:01:21.360,0:01:25.360
for now let's look at a worked example
0:01:23.200,0:01:26.960
where we show this explicitly
0:01:25.360,0:01:30.000
for the eigenstates of the
0:01:26.960,0:01:30.000
infinite potential well
0:01:32.880,0:01:35.759
who's a good boy
0:01:37.119,0:01:40.159
I've got a cameo from Geoffrey in the
0:01:38.840,0:01:42.880
background
0:01:40.159,0:01:44.720
all right so let's take a look at
0:01:42.880,0:01:46.479
the energy eigenstates of the infinite
0:01:44.720,0:01:48.399
potential well
0:01:46.479,0:01:50.320
another quick reminder as to what the
0:01:48.399,0:01:54.880
potential looks like
0:01:50.320,0:02:00.960
we have potential going from 0 to L
0:01:54.880,0:02:00.960
along x up to infinity here
0:02:04.240,0:02:07.360
and 0 within the well and we've seen
0:02:06.799,0:02:10.520
that
0:02:07.360,0:02:14.239
the normalized
0:02:10.520,0:02:14.239
eigenfunctions look like this
0:02:20.720,0:02:23.840
and this time we'd like to prove that
0:02:22.480,0:02:26.480
two
0:02:23.840,0:02:27.120
eigenfunctions are orthogonal to each
0:02:26.480,0:02:29.599
other
0:02:27.120,0:02:31.440
so what this means is that
0:02:29.599,0:02:32.000
the integral from minus infinity to infinity
0:02:31.440,0:02:42.080
of phi_n(x)*.phi_m(x) dx
0:02:38.160,0:02:45.200
equals delta_{nm}
0:02:42.080,0:02:48.560
which by definition equals
0:02:45.200,0:02:51.840
1 if n=m 0 if
0:02:48.560,0:02:51.840
n doesn't equal m
0:02:52.160,0:02:55.440
orthogonal of course implies
0:02:54.400,0:02:56.800
something
0:02:55.440,0:02:58.319
that there's some similarity to two
0:02:56.800,0:02:59.920
vectors being at 90 degrees and we'll
0:02:58.319,0:03:01.519
see when we study matrix mechanics that
0:02:59.920,0:03:05.200
there's actually a very close analogy
0:03:01.519,0:03:08.000
to that idea so let's
0:03:05.200,0:03:08.640
show this for the the eigenstates
0:03:08.000,0:03:10.720
of the
0:03:08.640,0:03:11.920
infinite well let's just substitute this
0:03:10.720,0:03:14.959
expression into here
0:03:11.920,0:03:17.040
so we get a 2/L out the front
0:03:14.959,0:03:19.760
the integral only goes from zero to L
0:03:17.040,0:03:22.959
because phi is 0 outside of that range
0:03:19.760,0:03:32.959
sin(n pi x/L)sin(m pi x/L)dx
0:03:30.959,0:03:34.720
okay we're going to need to use another
0:03:32.959,0:03:37.760
trigonometric identity
0:03:34.720,0:03:41.440
this time we need to use that
0:03:37.760,0:03:51.760
cos(A+B)=cos(A)cos(B)-sin(A)sin(B)
0:03:48.480,0:03:55.200
and therefore
0:03:51.760,0:04:04.720
cos(A-B)=cos(A)cos(B)+sin(A)sin(B)
0:04:06.080,0:04:10.080
okay so if we put A = n pi x/L
0:04:09.840,0:04:13.280
B = m pi x/L
0:04:10.080,0:04:16.160
we want to
0:04:13.280,0:04:17.680
add this one sorry subtract this one
0:04:16.160,0:04:18.880
from this one and that'll give us the
0:04:17.680,0:04:21.280
two sines
0:04:18.880,0:04:23.840
sorry you can't quite see that so we
0:04:21.280,0:04:25.360
want to subtract this expression from
0:04:23.840,0:04:28.800
this expression that will give us
0:04:25.360,0:04:32.240
2sin(A)sin(B) over here
0:04:28.800,0:04:33.520
and so overall we'll have so we'll
0:04:32.240,0:04:35.840
have a factor of two we bring out the
0:04:33.520,0:04:39.360
front so we have 1/L
0:04:35.840,0:04:42.639
integral from zero to L
0:04:39.360,0:04:48.000
cos(A-B) so it's
0:04:42.639,0:04:48.000
cos((n-m) pi x/L)
0:04:48.160,0:04:57.840
minus cos((n+m)pi x/L)
0:05:00.160,0:05:03.280
this integrates to
0:05:03.680,0:05:10.560
these will become sines
0:05:07.360,0:05:13.680
sin((n-m)pi x /L)/((n-m)pi)
0:05:13.680,0:05:18.920
and multiplied by L; minus
0:05:17.120,0:05:28.880
sin((n+m)pi x/L)/((n+m)pi)
0:05:24.720,0:05:32.160
multiply by L
0:05:28.880,0:05:33.199
between zero and L okay well the Ls
0:05:32.160,0:05:36.720
cancel so that's good
0:05:33.199,0:05:38.000
news when we stick in
0:05:36.720,0:05:40.479
sin(0)=0 so the zero
0:05:38.000,0:05:43.039
limit is always zero when we substitute
0:05:40.479,0:05:43.039
the L in
0:05:43.280,0:05:47.120
this one is always going to be zero
0:05:44.960,0:05:48.479
because n+m
0:05:47.120,0:05:50.400
for integer n and m is always an
0:05:48.479,0:05:52.400
integer as the sum of two integers is an
0:05:50.400,0:05:55.360
integer
0:05:52.400,0:05:56.479
and so this is ... substitute in
0:05:55.360,0:05:58.400
here these two cancel
0:05:56.479,0:06:00.080
so it's an integer times pi. Sine of that
0:05:58.400,0:06:03.199
is always zero so this one
0:06:00.080,0:06:04.080
disappears this one also almost
0:06:03.199,0:06:06.160
disappears
0:06:04.080,0:06:10.080
the zero limit disappears when we put L
0:06:06.160,0:06:12.560
in so we have sin((n-m)pi)
0:06:10.080,0:06:14.000
well n-m is also an integer for
0:06:12.560,0:06:16.240
integer n and m
0:06:14.000,0:06:18.319
the only problem is when n equals m this
0:06:16.240,0:06:21.280
is zero. sin(0)=0 -- good --
0:06:18.319,0:06:23.600
but n-m is also 0 on the bottom
0:06:21.280,0:06:25.199
and 0 divided by 0 is undefined
0:06:23.600,0:06:28.080
so let's substitute it in the only one
0:06:25.199,0:06:31.919
we need to worry about
0:06:28.080,0:06:38.639
sin((n-m)pi)/((n-m)pi)
0:06:36.800,0:06:40.400
so to work out what that is we use
0:06:38.639,0:06:41.280
l'Hopital's rule differentiate the top
0:06:40.400,0:06:42.960
and bottom
0:06:41.280,0:06:45.440
and we can differentiate them with
0:06:42.960,0:06:49.599
respect to, say, n-m
0:06:45.440,0:06:53.520
so this thing must equal
0:06:49.599,0:06:57.919
as we multiply this we get
0:06:53.520,0:06:57.919
cos((n-m)pi)
0:06:58.000,0:07:01.280
differentiate with respect to n-m
0:06:59.680,0:07:03.919
and we just get a pi on the bottom
0:07:01.280,0:07:04.560
cancel those this is evaluated at n
0:07:03.919,0:07:08.240
equals
0:07:04.560,0:07:10.319
m and so this thing equals one
0:07:08.240,0:07:12.319
and so we found that if we'd put
0:07:10.319,0:07:14.080
anything else anything other than n
0:07:12.319,0:07:15.520
equals m into here it would have
0:07:14.080,0:07:17.039
disappeared because if n
0:07:15.520,0:07:18.560
and m are different integers this is
0:07:17.039,0:07:21.280
non-zero this is zero
0:07:18.560,0:07:22.880
the whole thing is zero if n equals m it
0:07:21.280,0:07:23.919
evaluates to one and that's precisely
0:07:22.880,0:07:26.240
what we wanted to show
0:07:23.919,0:07:27.680
we've shown the orthogonality of
0:07:26.240,0:07:29.120
these eigenstates and in fact we've
0:07:27.680,0:07:31.440
shown that they're orthonormal because
0:07:29.120,0:07:35.520
we've already normalized the eigenstates
0:07:31.440,0:07:37.759
so the normalized eigenstates
0:07:35.520,0:07:39.840
of the states within the infinite
0:07:37.759,0:07:42.080
well
0:07:39.840,0:07:44.720
are orthonormal okay thanks for your
0:07:42.080,0:07:44.720
time
V3.5 Fourier decomposition
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
decomposing arbitrary functions (which match the boundary conditions) into weighted sums of energy eigenstates of the infinite potential well. Deriving the time dependence of arbitrary wave functions prepared in the well.
0:00:00.240,0:00:03.120
hello in this video we're going to take
0:00:02.560,0:00:06.160
a look
0:00:03.120,0:00:08.480
at fourier decomposition of functions
0:00:06.160,0:00:10.000
in the particular case of the infinite
0:00:08.480,0:00:13.280
potential well
0:00:10.000,0:00:16.400
so we showed in a previous video that
0:00:13.280,0:00:18.400
the eigenstates in the infinite
0:00:16.400,0:00:21.760
potential well the energy eigenstates
0:00:18.400,0:00:22.880
are all orthonormal in fact there's
0:00:21.760,0:00:24.800
a stronger condition
0:00:22.880,0:00:26.240
the energy eigenstates the infinite
0:00:24.800,0:00:30.240
potential well form what's called a
0:00:26.240,0:00:32.079
complete orthonormal basis
0:00:30.240,0:00:33.920
what we mean by this is that any
0:00:32.079,0:00:36.239
arbitrary function
0:00:33.920,0:00:36.960
of position within the well can be
0:00:36.239,0:00:39.280
written
0:00:36.960,0:00:40.399
as a sum of different energy
0:00:39.280,0:00:42.879
eigenstates
0:00:40.399,0:00:44.000
just clear the board okay so
0:00:42.879,0:00:48.320
mathematically what we're saying
0:00:44.000,0:00:50.800
is this: an arbitrary function
0:00:48.320,0:00:52.160
f(x) can be written as a sum from n
0:00:50.800,0:00:54.640
equals one to infinity
0:00:52.160,0:00:56.000
of phi_n(x) our energy eigenstates in the
0:00:54.640,0:00:57.920
infinite potential well
0:00:56.000,0:00:59.039
multiplied by potentially complex
0:00:57.920,0:01:02.239
coefficients
0:00:59.039,0:01:05.360
f_n. It's a form of Fourier analysis
0:01:02.239,0:01:07.280
in fact again it's not just
0:01:05.360,0:01:08.479
eigenstates for the infinite
0:01:07.280,0:01:10.240
potential well this
0:01:08.479,0:01:11.600
occurs in a much broader class of
0:01:10.240,0:01:13.280
cases in
0:01:11.600,0:01:15.600
quantum mechanics and we'll take a
0:01:13.280,0:01:17.439
closer look at the cases it applies to
0:01:15.600,0:01:19.600
when we come to matrix mechanics later
0:01:17.439,0:01:22.000
on
0:01:19.600,0:01:23.119
for now you can think of it in a very
0:01:22.000,0:01:26.080
close analogy
0:01:23.119,0:01:27.439
to expanding an arbitrary vector in
0:01:26.080,0:01:29.360
an n-dimensional vector space
0:01:27.439,0:01:30.640
in terms of the n basis vectors in the
0:01:29.360,0:01:33.040
space
0:01:30.640,0:01:33.840
we'll see that that analogy is indeed
0:01:33.040,0:01:36.640
very close
0:01:33.840,0:01:38.240
later on in the course for now in order
0:01:36.640,0:01:40.159
to make this useful we need a method of
0:01:38.240,0:01:42.320
solving for these complex coefficients
0:01:40.159,0:01:43.840
f_n and we can do that quite simply
0:01:42.320,0:01:46.399
using the orthonormality
0:01:43.840,0:01:47.040
of the energy eigenstates so we can
0:01:46.399,0:01:49.840
write the
0:01:47.040,0:01:50.159
following we can multiply from the left
0:01:49.840,0:01:53.200
by
0:01:50.159,0:01:54.960
phi_m*(x) and then integrate
0:01:53.200,0:01:56.640
from minus infinity to infinity in fact
0:01:54.960,0:01:59.680
this will only go over the
0:01:56.640,0:02:00.880
well because the phi_m are zero
0:01:59.680,0:02:02.159
outside as well
0:02:00.880,0:02:04.799
and when we do this on the right hand
0:02:02.159,0:02:04.799
side we have
0:02:04.880,0:02:08.239
the following so we've multiplied by
0:02:07.439,0:02:09.599
phi_m*(x)
0:02:08.239,0:02:11.360
in from the left and then we've
0:02:09.599,0:02:14.080
integrated dx
0:02:11.360,0:02:15.760
the phi_m*(x) can pass through the sum
0:02:14.080,0:02:16.480
and the integral dx can also pass
0:02:15.760,0:02:18.319
through the sum
0:02:16.480,0:02:20.400
that's because sums and integrals
0:02:18.319,0:02:22.879
commute
0:02:20.400,0:02:24.560
f_n are coefficients to be determined but
0:02:22.879,0:02:25.920
they're not a function of x so they can
0:02:24.560,0:02:29.680
come outside the integral
0:02:25.920,0:02:29.680
and so this quantity here
0:02:29.840,0:02:36.000
from our orthonormality condition is
0:02:32.480,0:02:39.040
just the kronecker delta
0:02:36.000,0:02:41.680
which is one if m equals n
0:02:39.040,0:02:42.480
and zero otherwise so when we sum over
0:02:41.680,0:02:44.319
n
0:02:42.480,0:02:45.760
the kronecker delta selects out the
0:02:44.319,0:02:48.640
case that n equals m
0:02:45.760,0:02:50.319
every other term is zero and so this
0:02:48.640,0:02:52.319
expression over here reduces to the
0:02:50.319,0:02:56.160
following
0:02:52.319,0:02:58.480
that is just f_m so
0:02:56.160,0:02:59.200
overall then if we just switch let's
0:02:58.480,0:03:01.040
swap these
0:02:59.200,0:03:02.480
the sides of these two things and switch
0:03:01.040,0:03:05.760
the label m to n
0:03:02.480,0:03:08.000
and we have the result
0:03:05.760,0:03:09.680
so we can write any arbitrary function
0:03:08.000,0:03:12.560
f(x) which
0:03:09.680,0:03:14.239
lies within the same
0:03:12.560,0:03:14.800
boundaries as the infinite potential
0:03:14.239,0:03:18.080
well
0:03:14.800,0:03:19.760
as a decomposition of energy eigenstates
0:03:18.080,0:03:21.599
in the potential well and the
0:03:19.760,0:03:23.519
coefficients we use in that expansion
0:03:21.599,0:03:25.040
we can determine from this simple
0:03:23.519,0:03:28.239
formula here
0:03:25.040,0:03:29.920
so this is already very useful it's
0:03:28.239,0:03:32.319
as useful as Fourier decomposition.
0:03:32.319,0:03:35.519
A particularly important use of this is
0:03:35.040,0:03:37.840
that
0:03:35.519,0:03:38.560
we can specify some starting wave
0:03:37.840,0:03:40.080
function
0:03:38.560,0:03:42.080
so it doesn't need to be an eigenstate
0:03:40.080,0:03:43.840
say we prepare a state
0:03:42.080,0:03:45.680
which is say a position eigenstate in
0:03:43.840,0:03:48.400
the well we identify the particle
0:03:45.680,0:03:48.879
at a particular position or we can put
0:03:48.879,0:03:52.640
in general quite different wave
0:03:51.599,0:03:54.000
functions in the well
0:03:52.640,0:03:56.799
depending on what types of measurements
0:03:54.000,0:03:59.280
we made and so on if we prepare them
0:03:56.799,0:04:00.560
or we can prepare
0:03:59.280,0:04:02.879
them by specifying
0:04:00.560,0:04:03.599
the amplitude at each point in the
0:04:02.879,0:04:05.840
well
0:04:03.599,0:04:06.720
but then we'd like to know how the wave
0:04:05.840,0:04:09.599
function
0:04:06.720,0:04:11.040
varies with time after that now the
0:04:09.599,0:04:12.879
time-dependent Schroedinger equation should
0:04:11.040,0:04:14.400
tell us the time evolution of any state
0:04:12.879,0:04:16.479
not just energy eigenstates
0:04:14.400,0:04:17.440
and this is how it does it because
0:04:16.479,0:04:20.479
remember
0:04:17.440,0:04:21.519
the time evolution of the energy
0:04:20.479,0:04:24.240
eigenstates
0:04:21.519,0:04:24.639
is trivial we know how to solve this and
0:04:24.240,0:04:26.800
so
0:04:24.639,0:04:28.080
we apply the same reasoning here we find
0:04:26.800,0:04:31.280
that starting off
0:04:28.080,0:04:33.840
in f(x) at time t=0
0:04:31.280,0:04:36.080
the result for f(x,t) at later times is
0:04:33.840,0:04:38.880
as follows
0:04:36.080,0:04:40.160
okay so specifying the wave function
0:04:38.880,0:04:41.840
at an initial instant
0:04:40.160,0:04:44.080
we know its behavior for all future
0:04:41.840,0:04:45.600
times because it must obey the time
0:04:44.080,0:04:47.919
dependent Schroedinger equation so it's
0:04:45.600,0:04:49.600
dictated
0:04:47.919,0:04:50.960
that is provided no measurement is made
0:04:49.600,0:04:54.800
to the system measurement
0:04:50.960,0:04:58.080
is very strange as we'll see shortly
0:04:54.800,0:05:00.880
so we specify our arbitrary
0:04:58.080,0:05:01.680
function of position its time evolution
0:05:00.880,0:05:03.759
we work out
0:05:01.680,0:05:05.840
by decomposing it into energy
0:05:03.759,0:05:07.600
eigenstates each of which
0:05:05.840,0:05:11.440
we know the time dependence of and it
0:05:07.600,0:05:13.280
takes this trivial form
0:05:11.440,0:05:15.840
the time dependence of the state itself
0:05:13.280,0:05:17.280
need no longer be trivial
0:05:15.840,0:05:18.880
because we're summing up different
0:05:17.280,0:05:21.120
energy eigenstates here
0:05:18.880,0:05:22.960
in fact this is a general expression of
0:05:21.120,0:05:23.520
quantum superposition which we'll take a
0:05:22.960,0:05:27.680
look at
0:05:23.520,0:05:27.680
in the next video thanks for your time
V4.1 Quantum superposition
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
linear combinations (superpositions) of solutions to the Schrödinger equation are also solutions; superpositions of stationary states need not themselves be stationary states; normalisation of superposed states; finding the probability of a superposition to give a particular outcome.
0:00:01.520,0:00:05.759
hello in this video we're going to take
0:00:03.360,0:00:08.880
a look at quantum superposition
0:00:05.759,0:00:12.800
the central story of the video is going
0:00:08.880,0:00:12.800
to be that quantum mechanics is linear
0:00:12.880,0:00:16.160
what we mean by that is that if two wave
0:00:15.599,0:00:18.800
functions
0:00:16.160,0:00:19.600
psi one and psi two which are independently
0:00:18.800,0:00:22.720
solutions
0:00:19.600,0:00:25.279
to the time-dependent Schroedinger equation
0:00:22.720,0:00:27.279
then any linear combination of them
0:00:25.279,0:00:29.279
is also a solution
0:00:27.279,0:00:30.560
that is wave functions of a linear
0:00:29.279,0:00:33.040
superposition
0:00:30.560,0:00:34.000
in general as we saw in a previous video
0:00:33.040,0:00:36.800
we can
0:00:34.000,0:00:36.800
write the following
0:00:37.360,0:00:41.600
if we can write some arbitrary wave
0:00:39.600,0:00:44.719
function down
0:00:41.600,0:00:45.760
we can decompose it in terms of energy
0:00:44.719,0:00:49.280
eigenstates
0:00:45.760,0:00:51.360
psi_n and
0:00:49.280,0:00:53.120
any linear combination like this will
0:00:51.360,0:00:55.280
also be a solution to the schrodinger
0:00:53.120,0:00:57.680
equation so it's not just a sum of two
0:00:55.280,0:00:59.199
it's a sum of any number of wave
0:00:57.680,0:01:02.800
functions which individually solve
0:00:59.199,0:01:02.800
the time dependent Schroedinger equation
0:01:02.879,0:01:06.640
let's clear the board just move this up
0:01:04.239,0:01:06.640
to the top
0:01:07.760,0:01:11.840
and just to reiterate if psi_n is a
0:01:10.720,0:01:12.960
solution to the time-dependent
0:01:11.840,0:01:14.960
Schroedinger equation
0:01:12.960,0:01:17.439
then any arbitrary function which is
0:01:14.960,0:01:20.400
written as a superposition a sum
0:01:17.439,0:01:21.200
of different psi_n weighted by
0:01:20.400,0:01:24.000
coefficients
0:01:21.200,0:01:24.479
f_n which are in general complex this is
0:01:24.000,0:01:29.040
also
0:01:24.479,0:01:31.200
a solution these are energy eigenstates
0:01:29.040,0:01:32.240
that we've been dealing with so far if
0:01:31.200,0:01:37.360
we perform
0:01:32.240,0:01:37.360
a measurement on f(x,t)
0:01:37.759,0:01:42.000
a measurement of energy this is where
0:01:40.960,0:01:42.960
things get a little bit strange in
0:01:42.000,0:01:45.600
quantum mechanics
0:01:42.960,0:01:48.159
we will always find exactly one of
0:01:45.600,0:01:51.200
the energy eigenstates
0:01:48.159,0:01:52.479
so let's write that down. A measurement
0:01:51.200,0:01:55.600
of the energy of
0:01:52.479,0:01:57.840
f(x,t) will reveal one energy E_n
0:01:55.600,0:01:59.439
so even though we prepare the state in a
0:01:57.840,0:02:01.360
superposition of different energy
0:01:59.439,0:02:03.280
eigenstates when we make a measurement
0:02:01.360,0:02:04.960
of the energy we only find one of the
0:02:03.280,0:02:08.080
energies
0:02:04.960,0:02:09.759
and as a result of that the wave
0:02:08.080,0:02:10.399
function will change as a result of that
0:02:09.759,0:02:12.879
measurement
0:02:10.399,0:02:14.640
it'll no longer be f(x,t) it will
0:02:12.879,0:02:18.560
then be the energy eigenstate
0:02:14.640,0:02:18.560
corresponding to that eigenenergy
0:02:19.120,0:02:22.800
after measuring eigenenergy eigenvalue
0:02:22.560,0:02:27.120
e_n
0:02:22.800,0:02:30.239
the state is psi_n(x,t)
0:02:27.120,0:02:30.239
and that's with certainty
0:02:30.560,0:02:34.160
whether you want to say that the state
0:02:32.000,0:02:36.000
has changed depends on your
0:02:34.160,0:02:38.080
interpretation of quantum mechanics
0:02:36.000,0:02:39.680
in the standard interpretation we tend
0:02:38.080,0:02:40.959
to teach at university, the Copenhagen
0:02:39.680,0:02:42.560
interpretation,
0:02:40.959,0:02:45.040
this process is called
0:02:42.560,0:02:45.040
'wave function collapse'
0:02:45.440,0:02:49.840
but this is an interpretational question
0:02:47.840,0:02:52.480
in the many worlds theory for example
0:02:49.840,0:02:53.680
wavefunction collapse does not exist
0:02:52.480,0:02:56.319
there is a different process for
0:02:53.680,0:02:59.440
explaining how a state prepared
0:02:56.319,0:03:02.080
in f(x,t) can, when measured
0:02:59.440,0:03:04.319
according to its energy, change into
0:03:02.080,0:03:08.560
psi_n(x,t) in fact it doesn't change
0:03:04.319,0:03:11.280
it just appears to change to us so
0:03:08.560,0:03:12.800
the probability when performing an
0:03:11.280,0:03:15.920
energy measurement on f
0:03:12.800,0:03:16.800
to find energy E_n and for the state
0:03:15.920,0:03:19.280
subsequently to be
0:03:16.800,0:03:20.480
psi_n is given by the modulus square of
0:03:19.280,0:03:22.879
the coefficient
0:03:20.480,0:03:23.519
assuming this is correctly normalized
0:03:22.879,0:03:25.040
which all
0:03:23.519,0:03:27.200
physical states in quantum mechanics
0:03:25.040,0:03:30.000
are
0:03:27.200,0:03:32.560
the probability to find the result E_n
0:03:30.000,0:03:35.680
in an energy measurement of f(x,t)
0:03:32.560,0:03:36.239
|f_n|^2 before the
0:03:35.680,0:03:38.640
measurement
0:03:36.239,0:03:39.280
after the measurement it will be in
0:03:38.640,0:03:40.799
state psi_n
0:03:39.280,0:03:42.799
and it'll have energy E_n with
0:03:40.799,0:03:43.840
probability one so the state really has
0:03:42.799,0:03:45.360
changed
0:03:43.840,0:03:47.200
we'll look at some of the more
0:03:45.360,0:03:48.159
philosophical interpretations
0:03:47.200,0:03:51.360
surrounding this
0:03:48.159,0:03:54.239
in a separate video for now let's
0:03:51.360,0:03:55.680
take a look at what this means for the
0:03:54.239,0:03:58.480
time dependence of states
0:03:55.680,0:04:01.599
let's just clear the board so let's take
0:03:58.480,0:04:04.799
a look at the time dependence
0:04:01.599,0:04:06.560
in the absence of measurement the
0:04:04.799,0:04:08.159
time dependence for state is dictated
0:04:06.560,0:04:09.120
entirely by the time dependent
0:04:08.159,0:04:11.680
Schroedinger equation
0:04:09.120,0:04:13.040
so we have that the state psi evolves
0:04:11.680,0:04:14.799
unitarily
0:04:13.040,0:04:18.400
according to the time-dependent Schroedinger
0:04:14.799,0:04:18.400
equation in the absence of measurement
0:04:18.720,0:04:23.199
the word unitarily here we'll see in
0:04:21.600,0:04:24.560
more detail what this means later on in
0:04:23.199,0:04:27.120
the course
0:04:24.560,0:04:28.960
but for now all you need to know is that
0:04:27.120,0:04:30.960
a unitary evolution of the wave function
0:04:28.960,0:04:32.160
simply preserves the normalization if
0:04:30.960,0:04:33.520
you start with a normalized wave
0:04:32.160,0:04:34.160
function which you must for a physical
0:04:33.520,0:04:36.720
state
0:04:34.160,0:04:37.280
it remains normalized for all subsequent
0:04:36.720,0:04:39.440
times
0:04:37.280,0:04:41.280
and this is built into the schrodinger
0:04:39.440,0:04:42.320
equation
0:04:41.280,0:04:44.240
okay so we'd like to look at the time
0:04:42.320,0:04:45.759
dependence of states remember if we have
0:04:44.240,0:04:47.440
an energy eigenstate
0:04:45.759,0:04:50.960
the probability density of that
0:04:47.440,0:04:53.280
eigenstate is time independent
0:04:50.960,0:04:55.680
so if this is our energy eigenstate
0:04:53.280,0:04:59.680
labeled with subscript n
0:04:55.680,0:05:03.280
then the probability density is this
0:04:59.680,0:05:04.880
so rho_n is modulus square of psi n
0:05:03.280,0:05:06.800
this is a function of time but the
0:05:04.880,0:05:09.039
result is completely equal to
0:05:06.800,0:05:10.320
phi_n only a function of
0:05:09.039,0:05:12.479
position x
0:05:10.320,0:05:13.759
and that's because of this form of the
0:05:12.479,0:05:15.600
time evolution
0:05:13.759,0:05:17.120
however when we take a quantum
0:05:15.600,0:05:18.000
superposition of two different energy
0:05:17.120,0:05:20.240
eigenstates
0:05:18.000,0:05:22.960
the result the resulting probability
0:05:20.240,0:05:25.280
density need not be time independent
0:05:22.960,0:05:26.880
so consider this state alpha psi one
0:05:25.280,0:05:28.960
plus b psi two where
0:05:26.880,0:05:30.160
psi one and psi two are different energy
0:05:28.960,0:05:33.919
eigen states
0:05:30.160,0:05:33.919
the probability density is this
0:05:34.320,0:05:38.080
it's equal to modulus psi squared again
0:05:36.720,0:05:41.440
and expanding
0:05:38.080,0:05:43.199
the product we find this result
0:05:41.440,0:05:45.280
and if we stick in the forms of the
0:05:43.199,0:05:48.160
energy eigenstates again
0:05:45.280,0:05:48.160
we find the result
0:05:48.560,0:05:52.160
so it takes this form from this term
0:05:51.759,0:05:54.160
here
0:05:52.160,0:05:55.360
psi one is time dependent but when we
0:05:54.160,0:05:57.199
take the modulus square of it because
0:05:55.360,0:05:58.960
it's an energy eigenstate the result is
0:05:57.199,0:06:01.600
the same as phi one squared
0:05:58.960,0:06:02.479
modulus which is time independent same
0:06:01.600,0:06:06.319
with phi two
0:06:02.479,0:06:09.600
but the cross-terms are in general
0:06:06.319,0:06:10.000
time dependent the result is still
0:06:09.600,0:06:11.680
real
0:06:10.000,0:06:13.199
because it's a probability density so it
0:06:11.680,0:06:14.560
better be real and we know it must be
0:06:13.199,0:06:15.199
real because we're taking the modulus
0:06:14.560,0:06:18.400
square
0:06:15.199,0:06:18.800
of some number which is always real
0:06:18.400,0:06:20.639
but
0:06:18.800,0:06:22.720
it may in general be time dependent even
0:06:20.639,0:06:26.000
though the states from which it's a sum
0:06:22.720,0:06:28.479
have separate time
0:06:26.000,0:06:30.560
independent probability
0:06:28.479,0:06:32.639
densities
0:06:30.560,0:06:34.560
okay now let's take a look at the
0:06:32.639,0:06:36.960
normalization of this superposition so
0:06:34.560,0:06:39.039
let's clear the board again
0:06:36.960,0:06:41.680
so look at normalization we'll pick the
0:06:39.039,0:06:43.600
same state as before
0:06:41.680,0:06:46.080
and remember that all physical states
0:06:43.600,0:06:48.639
must be normalized
0:06:46.080,0:06:50.319
which means that the modulus square of
0:06:48.639,0:06:50.880
the wave function integrated across all
0:06:50.319,0:06:53.840
of space
0:06:50.880,0:06:56.240
must equal one we ensure that our energy
0:06:53.840,0:06:58.000
eigenstates are normalized
0:06:56.240,0:06:59.599
and so this places a condition on the
0:06:58.000,0:07:00.880
possible alphas and betas we can take in
0:06:59.599,0:07:03.919
our superposition
0:07:00.880,0:07:07.280
in this case the condition is this
0:07:03.919,0:07:09.199
which equals again just expanding the
0:07:07.280,0:07:12.000
product into these four terms
0:07:09.199,0:07:13.680
but we know that not only are our energy
0:07:12.000,0:07:15.199
eigenstates normalized they're also
0:07:13.680,0:07:18.080
orthogonal to one another
0:07:15.199,0:07:19.440
so that condition by definition means
0:07:18.080,0:07:22.800
that this term is zero
0:07:19.440,0:07:23.280
because this is psi one and psi two
0:07:22.800,0:07:25.919
are
0:07:23.280,0:07:28.720
orthogonal so by definition this is zero
0:07:25.919,0:07:28.720
same with this term
0:07:28.800,0:07:32.000
and we also have that these two states
0:07:30.880,0:07:33.759
are normalized
0:07:32.000,0:07:36.160
so putting this together we find the
0:07:33.759,0:07:36.160
following
0:07:36.240,0:07:41.039
that is one equals the modulus of alpha
0:07:38.319,0:07:42.960
squared plus the modulus of beta squared
0:07:41.039,0:07:44.080
so this gives us our normalization that
0:07:42.960,0:07:47.120
must appear on
0:07:44.080,0:07:48.720
this wave function as the following that
0:07:47.120,0:07:49.440
is we now have a properly normalized
0:07:48.720,0:07:53.919
wave function
0:07:49.440,0:07:56.000
psi for arbitrary complex alpha and beta
0:07:53.919,0:07:57.280
if we perform an energy measurement on
0:07:56.000,0:07:59.520
psi
0:07:57.280,0:08:01.039
it's not itself in energy eigenstates we
0:07:59.520,0:08:03.759
won't certainly get
0:08:01.039,0:08:05.199
any particular energy we can say for
0:08:03.759,0:08:08.319
certainty that we won't get
0:08:05.199,0:08:10.479
any result other than E_1 or E_2 because
0:08:08.319,0:08:12.240
the amplitude for any other energy
0:08:10.479,0:08:13.520
eigenstate is zero
0:08:12.240,0:08:16.319
it didn't need to be but it is in this
0:08:13.520,0:08:17.840
particular choice
0:08:16.319,0:08:20.720
and the probability that we'll find
0:08:17.840,0:08:23.680
energy E_1 is given by
0:08:20.720,0:08:25.360
the modulus of alpha squared divided by
0:08:23.680,0:08:26.000
the normalization which would be this
0:08:25.360,0:08:27.840
thing squared
0:08:26.000,0:08:29.120
so it'll be modulus alpha squared
0:08:27.840,0:08:32.320
divided by alpha modulus
0:08:29.120,0:08:34.080
squared plus modulus beta squared and
0:08:32.320,0:08:35.039
similarly the probability for finding
0:08:34.080,0:08:36.880
energy E_2
0:08:35.039,0:08:38.560
will be the modulus of beta squared
0:08:36.880,0:08:40.640
divided by the normalization
0:08:38.560,0:08:41.919
so this guarantees that the total
0:08:40.640,0:08:44.320
probability to find
0:08:41.919,0:08:46.480
the particle in some energy is equal to
0:08:44.320,0:08:48.160
one
0:08:46.480,0:08:49.760
as always when we carry out our
0:08:48.160,0:08:51.279
normalization notice that the
0:08:49.760,0:08:52.959
normalization condition
0:08:51.279,0:08:54.640
only places a constraint on the
0:08:52.959,0:08:58.000
magnitude not the phase
0:08:54.640,0:08:59.760
so we're left with a global phase (which
0:08:58.000,0:09:03.200
is again ambiguous) out the front
0:08:59.760,0:09:04.720
of this wave function but the relative
0:09:03.200,0:09:06.720
phase between these two
0:09:04.720,0:09:08.720
contributions to the superposition is
0:09:06.720,0:09:10.959
important so that would change things
0:09:08.720,0:09:12.880
but there's a global complex phase at
0:09:10.959,0:09:14.560
the front of this which is arbitrary
0:09:12.880,0:09:16.399
because the global phase in quantum
0:09:14.560,0:09:19.680
mechanics is unobservable
0:09:16.399,0:09:19.680
okay thanks for your time
V4.2 The finite potential well
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
establishing properties of the bound states within the finite potential well.
0:00:01.040,0:00:03.439
hello in this video we're going to take
0:00:02.960,0:00:05.520
a look
0:00:03.439,0:00:06.640
at the finite potential well we've
0:00:05.520,0:00:07.520
looked previously at the infinite
0:00:06.640,0:00:08.960
potential well
0:00:07.520,0:00:10.960
now we're going to bring that potential
0:00:08.960,0:00:14.400
down to a finite value so
0:00:10.960,0:00:17.920
the potential is as follows
0:00:14.400,0:00:19.760
so zero within a region of x as before
0:00:17.920,0:00:21.039
and V_0 otherwise where V_0 is
0:00:19.760,0:00:22.720
no longer infinity
0:00:21.039,0:00:24.560
this time we've set it up so that the
0:00:22.720,0:00:27.599
well is symmetric about zero
0:00:24.560,0:00:28.640
it just makes the maths slightly
0:00:27.599,0:00:31.679
simpler to work with
0:00:28.640,0:00:34.399
let's draw it
0:00:31.679,0:00:34.880
so it's symmetric about zero and it has
0:00:34.399,0:00:39.280
height V_0
0:00:34.880,0:00:41.680
and this is zero down here
0:00:39.280,0:00:43.760
now let's try to guess what the
0:00:41.680,0:00:47.200
solutions look like
0:00:43.760,0:00:48.879
so down at the bottom of the well
0:00:47.200,0:00:49.920
we'll have some bound state down here
0:00:48.879,0:00:50.960
just like we did in the infinite
0:00:49.920,0:00:52.960
potential well
0:00:50.960,0:00:54.960
I'm going to sketch it in this slightly
0:00:52.960,0:00:58.079
dodgy way of drawing the wave functions
0:00:54.960,0:00:59.199
onto the potential plot at a snapshot in
0:00:58.079,0:01:02.000
time
0:00:59.199,0:01:03.680
so before the lowest energy state looked
0:01:02.000,0:01:06.559
like this in fact let's draw that
0:01:03.680,0:01:06.559
in a different colour
0:01:09.280,0:01:14.320
so the solution to the infinite
0:01:10.640,0:01:14.320
potential well looked like this
0:01:14.479,0:01:17.600
we're going to have something like that
0:01:15.840,0:01:19.360
but there's no longer a requirement that
0:01:17.600,0:01:20.720
the wave function vanishes at this point
0:01:19.360,0:01:22.000
because remember the wave function only
0:01:20.720,0:01:24.000
has to vanish in regions where the
0:01:22.000,0:01:27.040
potential is infinity
0:01:24.000,0:01:28.479
so using this as a motivation
0:01:27.040,0:01:31.840
we can guess that the lowest energy
0:01:28.479,0:01:31.840
state might look something like this
0:01:35.920,0:01:40.000
that is it takes the form of standing
0:01:39.119,0:01:41.680
waves within the
0:01:40.000,0:01:43.360
well but it doesn't need to vanish at
0:01:41.680,0:01:43.840
the edge of the well and outside of the
0:01:43.360,0:01:45.600
well
0:01:43.840,0:01:48.000
it isn't zero it's exponentially
0:01:45.600,0:01:49.600
decreasing remember our solutions both
0:01:48.000,0:01:51.920
to the infinite potential well here
0:01:49.600,0:01:52.960
and also to our scattering from an
0:01:51.920,0:01:56.719
infinitely long
0:01:52.960,0:01:59.520
but finite height potential step
0:01:56.719,0:02:00.399
so that's the lowest energy state we
0:01:59.520,0:02:02.719
might guess
0:02:00.399,0:02:04.079
similarly that if we have another bound
0:02:02.719,0:02:07.280
state up here
0:02:04.079,0:02:07.280
it might look something like this
0:02:07.520,0:02:11.599
where again this is a snapshot in time
0:02:09.200,0:02:14.800
of the real part of the wavefunction psi
0:02:11.599,0:02:17.520
and this will again come off to
0:02:14.800,0:02:17.520
zero like this
0:02:19.040,0:02:22.480
and we might have higher energy bound
0:02:21.040,0:02:23.760
states in the well there's no
0:02:22.480,0:02:25.200
requirement for there to be an infinite
0:02:23.760,0:02:26.720
number of bound states which there was
0:02:25.200,0:02:29.040
in the infinite potential well
0:02:26.720,0:02:30.800
we can only fit some number in before
0:02:29.040,0:02:31.920
the energy of the states is higher than
0:02:30.800,0:02:33.680
that of the well
0:02:31.920,0:02:36.239
and when we do that we then expect the
0:02:33.680,0:02:37.599
solutions to be plane waves there will
0:02:36.239,0:02:39.519
be some kind of boundary condition going
0:02:37.599,0:02:42.160
on here because of the edges of the
0:02:39.519,0:02:43.599
well but in general we'll have a
0:02:42.160,0:02:45.680
continuum of different
0:02:43.599,0:02:47.040
plane wave states above the well so
0:02:45.680,0:02:48.640
there's an infinite number of these
0:02:47.040,0:02:51.040
there's only a finite number of states
0:02:48.640,0:02:53.519
trapped within the well
0:02:51.040,0:02:54.319
taking a look at the forms in the
0:02:53.519,0:02:55.599
different regions
0:02:54.319,0:02:57.920
we can guess that we'll have standing
0:02:55.599,0:02:59.680
waves in here remember our general
0:02:57.920,0:03:00.959
types of wave solution are either plane
0:02:59.680,0:03:02.400
waves other
0:03:00.959,0:03:04.720
types of solution for regions with
0:03:02.400,0:03:06.720
constant potential are the plane waves
0:03:04.720,0:03:08.640
which we have up here they're
0:03:06.720,0:03:09.440
standing waves a form of plane wave but
0:03:08.640,0:03:10.959
where we have
0:03:09.440,0:03:12.480
equal contributions from left- and right-
0:03:10.959,0:03:13.519
going waves which we expect within the
0:03:12.480,0:03:15.440
well
0:03:13.519,0:03:17.680
and they can be evanescent waves
0:03:15.440,0:03:20.080
exponentially increasing or decreasing
0:03:17.680,0:03:20.800
and note that over here our our physical
0:03:20.080,0:03:22.400
guess
0:03:20.800,0:03:26.000
was that we have exponentially
0:03:22.400,0:03:29.760
increasing solutions in this region
0:03:26.000,0:03:32.959
and a decreasing solution over here
0:03:29.760,0:03:34.720
okay so let's write down the forms of
0:03:32.959,0:03:36.000
those wave functions and substitute in
0:03:34.720,0:03:38.319
the boundary conditions on the next
0:03:36.000,0:03:42.159
board
0:03:38.319,0:03:42.159
sorry didn't erase it one more go
0:03:43.599,0:03:47.840
sorry the board's playing up
0:03:48.159,0:03:51.599
oh sorry I must have it set to change my
0:03:50.879,0:03:55.360
clothes
0:03:51.599,0:03:58.640
right oh hi okay got it
0:03:55.360,0:03:59.760
got it okay all right so wave functions
0:03:58.640,0:04:01.360
in the different regions
0:03:59.760,0:04:03.120
it depends on whether the energy is
0:04:01.360,0:04:05.120
greater than or less than V_0
0:04:03.120,0:04:06.480
if it's greater than V_0 we're
0:04:05.120,0:04:07.519
just back to plane wave solutions in
0:04:06.480,0:04:09.200
all three regions
0:04:07.519,0:04:10.640
and we're solving exactly the same
0:04:09.200,0:04:12.480
problem as
0:04:10.640,0:04:14.159
the scattering over the top of a
0:04:12.480,0:04:14.959
potential barrier which you've seen in a
0:04:14.159,0:04:17.280
previous video
0:04:14.959,0:04:18.000
so let's only consider the bound states
0:04:17.280,0:04:20.400
which lie
0:04:18.000,0:04:21.120
with energy less than V_0 in that
0:04:20.400,0:04:22.240
case
0:04:21.120,0:04:24.240
we have the following results in
0:04:22.240,0:04:26.880
the different regions
0:04:24.240,0:04:27.680
so let's call it region one; x is to the
0:04:26.880,0:04:30.479
left of the
0:04:27.680,0:04:31.520
well so less than -L/2 we
0:04:30.479,0:04:34.400
have phi one
0:04:31.520,0:04:34.880
equals unknown coefficient a
0:04:34.400,0:04:37.919
times e^(i kappa x)
0:04:34.880,0:04:40.080
it must be exponentially increasing
0:04:37.919,0:04:42.960
in order to die off at x equals minus
0:04:40.080,0:04:46.240
infinity rather than blow up
0:04:42.960,0:04:48.960
similarly in region 3
0:04:46.240,0:04:50.720
that is x>L/2. In
0:04:48.960,0:04:53.040
region 3 you must have the form
0:04:50.720,0:04:54.000
unknown constant d times e^(-kappa x)
0:04:53.040,0:04:56.000
where
0:04:54.000,0:04:57.680
kappa is the same kappa as appeared in
0:04:56.000,0:04:58.320
region 1 because the potentials are the
0:04:57.680,0:05:00.800
same
0:04:58.320,0:05:01.520
both equal to V_0 that is both of
0:05:00.800,0:05:02.800
these
0:05:01.520,0:05:04.560
when substituted into the time
0:05:02.800,0:05:06.960
independent Schroedinger equation gives
0:05:04.560,0:05:08.880
the form
0:05:06.960,0:05:10.800
minus h bar squared kappa squared over
0:05:08.880,0:05:13.520
2m plus V_0
0:05:10.800,0:05:14.240
and this ensures that we can have real
0:05:13.520,0:05:16.080
kappa
0:05:14.240,0:05:18.479
for E; the bra
0:02:07.280,0:02:11.440
the Hermitian conjugate of |v> and let's
0:02:10.239,0:02:15.120
let's define a bra
0:02:11.440,0:02:17.440
__ we have the
0:02:40.879,0:02:46.000
following
0:02:42.959,0:02:47.920
the sum of u_n*.v_n so the elements
0:02:46.000,0:02:50.879
we're taking the complex conjugate this
0:02:47.920,0:02:53.440
one and the result here must be
0:02:50.879,0:02:54.239
a one by one matrix which is just a
0:02:53.440,0:02:57.519
complex
0:02:54.239,0:03:00.879
scalar so the bracket forms
0:02:57.519,0:03:03.599
complex scalars
0:03:00.879,0:03:05.519
so compare this to our usual or the
0:03:03.599,0:03:06.959
perhaps more familiar vector notation
0:03:05.519,0:03:08.800
where we just underline or we could use
0:03:06.959,0:03:12.159
bold and so on
0:03:08.800,0:03:15.040
but if we take u dagger dot v
0:03:12.159,0:03:16.400
where the v dot is the dot product or
0:03:15.040,0:03:17.840
the inner product
0:03:16.400,0:03:19.519
then we would also get this complex
0:03:17.840,0:03:20.640
scalar we get the same thing so that's
0:03:19.519,0:03:22.239
all we're doing here it's just a
0:03:20.640,0:03:23.599
different notation
0:03:22.239,0:03:25.200
a particular convenience of this
0:03:23.599,0:03:27.120
notation though is that if we want to
0:03:25.200,0:03:29.519
look at the complex conjugate of this
0:03:27.120,0:03:30.720
so remember this bracket is a
0:03:29.519,0:03:35.040
complex number
0:03:30.720,0:03:37.440
a complex scalar if we take the
0:03:35.040,0:03:39.680
complex conjugate of that we just get
0:03:37.440,0:03:42.159
the following
0:03:39.680,0:03:42.879
so the complex conjugate of u inner
0:03:42.159,0:03:45.760
product v
0:03:42.879,0:03:46.560
or bracket ____ is equal to v inner
0:03:45.760,0:03:48.239
product u
0:03:46.560,0:03:49.680
and you can just check this explicitly
0:03:48.239,0:03:51.280
in terms of the elements
0:03:49.680,0:03:53.840
so that's a nice convenience of this
0:03:51.280,0:03:54.319
notation this also tells us that if
0:03:53.840,0:03:57.439
we take
0:03:54.319,0:03:58.319
the inner product of v with itself we
0:03:57.439,0:04:00.720
must get
0:03:58.319,0:04:03.519
the sum over v_n*.v_n and that's
0:04:00.720,0:04:07.120
nothing other than
0:04:03.519,0:04:08.879
we just get the norm of the vector v
0:04:07.120,0:04:10.560
squared where norm is just a
0:04:08.879,0:04:12.720
slight generalization of the
0:04:10.560,0:04:14.000
length of the vector so this makes
0:04:12.720,0:04:18.239
sense if we take v
0:04:14.000,0:04:20.239
dot v for for real vectors we expect
0:04:18.239,0:04:21.919
to get the length squared or the
0:04:20.239,0:04:25.120
modulus squared
0:04:21.919,0:04:27.840
okay so we have
0:04:25.120,0:04:30.320
a complex vector space with an inner
0:04:27.840,0:04:31.440
product on it or a dot product
0:04:30.320,0:04:35.440
and actually we have the following
0:04:31.440,0:04:37.680
definition so our definition
0:04:35.440,0:04:38.880
a complex linear vector space endowed
0:04:37.680,0:04:41.600
with an inner product
0:04:38.880,0:04:43.120
in which all vectors are normalizable is
0:04:41.600,0:04:43.759
an example of what's called a Hilbert
0:04:43.120,0:04:44.880
space
0:04:43.759,0:04:47.120
so when we say that the vectors are
0:04:44.880,0:04:51.040
normalizable this means that their
0:04:47.120,0:04:53.360
norms squared are all finite
0:04:51.040,0:04:55.120
so this is the relevant to these complex
0:04:53.360,0:04:56.560
vector spaces to quantum mechanics
0:04:55.120,0:04:59.600
although we've dealt with wave functions
0:04:56.560,0:05:01.919
so far we'll see in an upcoming video
0:04:59.600,0:05:03.360
how those fit into this scheme but in
0:05:01.919,0:05:04.320
complete generality we can say that
0:05:03.360,0:05:06.479
in quantum mechanics
0:05:04.320,0:05:09.280
the states live in
0:05:06.479,0:05:09.280
Hilbert spaces
0:05:09.520,0:05:14.960
okay all right so let's take a look at
0:05:12.400,0:05:18.000
matrices acting on our vectors
0:05:14.960,0:05:18.800
okay so if we have a matrix M acting on
0:05:18.000,0:05:22.080
a vector
0:05:18.800,0:05:24.880
|u> in general we expect to get
0:05:22.080,0:05:27.520
some other vector let's call it |v>.
0:05:24.880,0:05:30.400
M is an n by n matrix
0:05:27.520,0:05:31.280
|u> being a vector must be an n by one
0:05:30.400,0:05:33.199
matrix
0:05:31.280,0:05:36.160
and an n by n matrix acting on n by one
0:05:33.199,0:05:38.320
matrix gives us an n by one matrix
0:05:36.160,0:05:40.560
we just cancelled it out to the middle
0:05:38.320,0:05:41.280
and so this works out so a matrix acting
0:05:40.560,0:05:43.120
on a vector
0:05:41.280,0:05:46.320
gives us another vector in this
0:05:43.120,0:05:48.400
complex space so that's good news
0:05:46.320,0:05:50.240
now consider the inner product let's
0:05:48.400,0:05:53.600
take another vector |w>
0:05:50.240,0:05:56.400
act it on M|u>
0:05:53.600,0:05:57.680
well this thing must by definition
0:05:56.400,0:06:01.919
then equal
0:05:57.680,0:06:07.360
|w> inner product |v> because M|u> is just |v>
0:06:01.919,0:06:10.080
and so this thing is a complex scalar
0:06:07.360,0:06:11.759
so that's also good news we can act
0:06:10.080,0:06:13.199
matrices on our vectors and
0:06:11.759,0:06:16.160
we can take inner products and
0:06:13.199,0:06:18.840
everything works out as we'd expect
0:06:16.160,0:06:20.720
okay so we've taken a look at the inner
0:06:18.840,0:06:23.680
product we saw
0:06:20.720,0:06:25.360
that so we write the complex
0:06:23.680,0:06:26.240
conjugate row vector here conjugate
0:06:25.360,0:06:28.240
transpose
0:06:26.240,0:06:29.280
multiplied by the vector gives us a
0:06:28.240,0:06:31.759
complex scalar
0:06:29.280,0:06:34.319
which is the inner product how about
0:06:31.759,0:06:37.280
this object
0:06:34.319,0:06:37.680
so i've just written the ket |v> on the
0:06:37.280,0:06:42.240
left
0:06:37.680,0:06:45.759
of the bra ____ where i
0:07:44.560,0:07:50.319
ranges from 1 to n in the n-dimensional
0:07:47.919,0:07:50.319
space
0:07:50.560,0:07:54.160
and they're defined by the following
0:07:52.400,0:07:57.280
fact
0:07:54.160,0:07:59.199
the inner product between |e_i> and |e_j>
0:07:57.280,0:08:00.560
where remember this is defined as the
0:07:59.199,0:08:03.599
hermitian conjugate
0:08:00.560,0:08:05.520
of vector |e_i> or ket |e_i>
0:08:03.599,0:08:06.879
this inner product is the kronecker
0:08:05.520,0:08:08.560
delta, \delta_{ij}
0:08:06.879,0:08:11.360
which is defined to be 0 if i doesn't
0:08:08.560,0:08:14.479
equal j and 1 if i does equal j
0:08:11.360,0:08:17.520
so we can for example sandwich a matrix
0:08:14.479,0:08:20.160
between two basis vectors
0:08:17.520,0:08:22.960
and we'll simply select out element i j
0:08:20.160,0:08:24.960
of the matrix
0:08:22.960,0:08:26.879
similarly with vectors we can take any
0:08:24.960,0:08:29.919
vector we like and decompose it
0:08:26.879,0:08:30.879
into any complete orthonormal basis such
0:08:29.919,0:08:33.680
as this
0:08:30.879,0:08:35.680
so for example we can write this that is
0:08:33.680,0:08:38.399
we can write any vector
0:08:35.680,0:08:40.080
as a sum over the basis vectors |e_i>
0:08:38.399,0:08:42.560
multiplied by coefficients
0:08:40.080,0:08:43.279
where |v_i> is the projection of the
0:08:42.560,0:08:46.640
vector v
0:08:43.279,0:08:49.120
along the basis direction |e_i> so
0:08:46.640,0:08:49.760
that's completely general but then if
0:08:49.120,0:08:50.959
you
0:08:49.760,0:08:52.800
think about what we're saying
0:08:50.959,0:08:53.680
this |v_i> is we want it to be the
0:08:52.800,0:08:56.800
projection
0:08:53.680,0:08:58.480
of vector v along direction |e_i>
0:08:56.800,0:09:01.600
but in our Dirac notation that's
0:08:58.480,0:09:03.920
nothing other than the following
0:09:01.600,0:09:05.760
that is |v_i> is given by take the
0:09:03.920,0:09:08.000
vector |v> and we project it along
0:09:05.760,0:09:10.399
the direction |e_i> remember that an inner
0:09:08.000,0:09:12.000
product is the projection of one vector
0:09:10.399,0:09:14.080
along another
0:09:12.000,0:09:15.040
so another way to rewrite this which
0:09:14.080,0:09:18.240
looks even neater
0:09:15.040,0:09:19.839
in Dirac notation is this
0:09:18.240,0:09:21.680
that is all i've brought is i've brought
0:09:19.839,0:09:23.920
the ket |e_i> over to the left
0:09:21.680,0:09:25.519
this remember is a complex scalar so it
0:09:23.920,0:09:27.040
can just pull through here
0:09:25.519,0:09:28.959
and so you see that you have the
0:09:27.040,0:09:32.399
outer product of the |e_i> with itself
0:09:28.959,0:09:33.839
acting on the vector |v> just as a
0:09:32.399,0:09:36.720
quick mention of notation
0:09:33.839,0:09:39.200
when i draw the outer products between
0:09:36.720,0:09:42.320
say u and v
0:09:39.200,0:09:42.320
really it's ket
0:09:42.720,0:09:48.320
bra like this but it's much easier to
0:09:45.440,0:09:49.519
actually write it down as follows
0:09:48.320,0:09:51.200
just in terms of how you actually write
0:09:49.519,0:09:52.240
it if you draw a cross (X) I think it's
0:09:51.200,0:09:54.160
indistinguishable
0:09:52.240,0:09:55.839
but i'm not writing a cross here i'm
0:09:54.160,0:09:59.120
doing a ket followed by
0:09:55.839,0:10:02.000
a bra okay so
0:09:59.120,0:10:02.720
in order to prove a very useful
0:10:02.000,0:10:06.640
relation
0:10:02.720,0:10:06.640
we need to use the following theorem
0:10:07.600,0:10:13.440
if we have two matrices A and B and
0:10:11.040,0:10:15.519
the inner product well if a acts on v
0:10:13.440,0:10:17.680
and we take the inner product with u
0:10:15.519,0:10:19.279
and that thing is equal to u inner
0:10:17.680,0:10:22.480
product B acting on v
0:10:19.279,0:10:24.399
for all u and v arbitrarily then
0:10:22.480,0:10:25.600
that means that the matrix A is equal to
0:10:24.399,0:10:26.560
the matrix B
0:10:25.600,0:10:28.800
this should hopefully make some
0:10:26.560,0:10:31.680
intuitive sense
0:10:28.800,0:10:34.880
so if we use this theorem we can
0:10:31.680,0:10:36.880
prove the following very nice result
0:10:34.880,0:10:38.560
rewriting our vector v written out in
0:10:36.880,0:10:40.560
its basis |e_i>
0:10:38.560,0:10:42.640
rewriting the vector |v> projected into
0:10:40.560,0:10:46.210
the basis |e_i>
0:10:42.640,0:10:47.440
we can act from the left with some bra ____ (or |phi_n> if we use the older
0:09:56.160,0:09:59.360
notation)
0:09:57.600,0:10:00.640
and you can check straightforwardly that
0:09:59.360,0:10:01.920
this matrix H
0:10:00.640,0:10:04.000
obeys all the properties you'd like it
0:10:01.920,0:10:07.040
to obey for example it returns
0:10:04.000,0:10:07.040
the right eigenvalue
0:10:07.279,0:10:10.800
that is we can act this whole object
0:10:10.320,0:10:13.680
onto
0:10:10.800,0:10:14.800
m and the m is is happy going in and out
0:10:13.680,0:10:17.760
of the sum
0:10:14.800,0:10:18.800
but this object here by definition
0:10:17.760,0:10:21.519
remember that we have
0:10:18.800,0:10:22.560
a complete orthonormal basis for our
0:10:21.519,0:10:25.279
eigenvectors
0:10:22.560,0:10:25.839
of any Hermitian matrix but in particular
0:10:25.279,0:10:28.160
of the
0:10:25.839,0:10:30.880
hamiltonian this thing is just a
0:10:28.160,0:10:33.040
Kronecker delta
0:10:30.880,0:10:34.000
defined to be one when n equals m and
0:10:33.040,0:10:36.399
zero otherwise
0:10:34.000,0:10:37.040
so it selects from the sum the case that
0:10:37.040,0:10:41.519
oh sorry i've written this sum over i
0:10:39.839,0:10:44.000
there should have been a sum over
0:10:41.519,0:10:44.959
n in this case it's selected from the
0:10:44.000,0:10:49.440
sum the case
0:10:44.959,0:10:49.440
n equals m and so we just get the result
0:10:49.600,0:10:53.040
E_m acting on |m> but that's
0:10:51.760,0:10:54.800
precisely our
0:10:53.040,0:10:56.399
time-independent Schroedinger equation so
0:10:54.800,0:10:58.000
that works and we can also check that
0:10:56.399,0:11:02.000
this matrix is by definition
0:10:58.000,0:11:04.320
Hermitian conjugate of H
0:11:02.000,0:11:06.320
we complex conjugate the energy but the
0:11:04.320,0:11:08.640
energies are real so this is equal to E_n
0:11:06.320,0:11:10.800
nothing changes and the Hermitian
0:11:08.640,0:11:14.079
conjugate of the matrix formed by
0:11:10.800,0:11:16.800
an outer product n remember this is
0:11:14.079,0:11:17.519
a NxN matrix well clearly
0:11:16.800,0:11:19.519
it's just
0:11:17.519,0:11:20.720
you flip them around but you just get
0:11:19.519,0:11:22.640
an outer product n again
0:11:20.720,0:11:23.839
so you get exactly the same thing so
0:11:22.640,0:11:25.760
this thing by construction is always
0:11:23.839,0:11:29.680
Hermitian
0:11:25.760,0:11:32.959
and we find that H dagger equals H
0:11:29.680,0:11:34.000
and so H is Hermitian okay so there's a
0:11:32.959,0:11:36.000
simple example
0:11:34.000,0:11:39.680
of putting some of these theorems to use
0:11:36.000,0:11:39.680
all right thank you for your time
V5.3a Spin-1/2 (part I)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
introduction to the spin (intrinsic angular momentum) of a particle; experimental results regarding spin-1/2 particles deduced from the Stern Gerlach experiment. Continued in video V5.3b.
0:00:01.839,0:00:04.240
hello in this video we're going to take
0:00:03.760,0:00:06.879
a look
0:00:04.240,0:00:08.080
at spin one half it's a particularly
0:00:06.879,0:00:10.080
simple example
0:00:08.080,0:00:11.200
of a finite dimensional hilbert space
0:00:10.080,0:00:14.480
which we've encountered
0:00:11.200,0:00:17.199
in the previous two videos so spin
0:00:14.480,0:00:19.279
is a quantum property which is also
0:00:17.199,0:00:22.880
known as
0:00:19.279,0:00:24.480
intrinsic angular momentum so if we
0:00:22.880,0:00:26.960
think kind of classically for a minute
0:00:24.480,0:00:29.119
and think of the electron
0:00:26.960,0:00:31.519
in the atom as like the earth orbiting
0:00:29.119,0:00:33.360
the sun so kind of orbiting like this
0:00:31.519,0:00:34.800
well that has angular momentum but
0:00:33.360,0:00:35.920
you can also have the earth spinning on
0:00:34.800,0:00:37.280
its own axis
0:00:35.920,0:00:39.120
and that's what intrinsic angular
0:00:37.280,0:00:40.480
momentum is like okay slightly spinning
0:00:39.120,0:00:41.760
on its own axis
0:00:40.480,0:00:43.440
but things are obviously going to have
0:00:41.760,0:00:45.120
to get a little bit weird because while
0:00:43.440,0:00:47.039
the earth has a finite radius
0:00:45.120,0:00:50.320
the radius of the electron as far as we
0:00:47.039,0:00:53.360
can tell is effectively zero
0:00:50.320,0:00:55.520
so we can identify properties of the
0:00:53.360,0:00:56.840
spin of an electron using what's called
0:00:55.520,0:00:59.280
a Stern-Gerlach
0:00:56.840,0:01:00.320
apparatus so there's an experiment that
0:00:59.280,0:01:02.719
was carried out
0:01:00.320,0:01:05.519
in 1922 originally i believe and it
0:01:02.719,0:01:08.640
looks something like this
0:01:05.519,0:01:10.400
that is a very a large magnet
0:01:08.640,0:01:11.680
we've got the north pole up here in the
0:01:10.400,0:01:13.680
south pole up here this is somewhat
0:01:11.680,0:01:15.360
schematic
0:01:13.680,0:01:17.759
and in particular the apparatus
0:01:15.360,0:01:18.560
generates a large field gradient in this
0:01:17.759,0:01:22.320
direction
0:01:18.560,0:01:23.759
so we have a gradient of the magnetic
0:01:22.320,0:01:26.799
field B
0:01:23.759,0:01:27.520
directed in this direction now the
0:01:26.799,0:01:29.040
electron
0:01:27.520,0:01:30.880
if it has this intrinsic angular
0:01:29.040,0:01:33.040
momentum if it's spinning well
0:01:30.880,0:01:35.439
it's got an electric charge so we
0:01:33.040,0:01:37.759
might expect it to have a magnetic field
0:01:35.439,0:01:38.880
and we can shoot a beam of electrons
0:01:37.759,0:01:40.960
down through here
0:01:38.880,0:01:43.280
i think this was done by just taking a
0:01:40.960,0:01:46.240
heated element and it will emit
0:01:43.280,0:01:46.240
electrons naturally [actually Silver atoms, sorry!]
0:01:49.600,0:01:53.280
and we can shoot our beam of electrons [Ag atoms]
0:01:51.040,0:01:56.799
down through here and they'll deflect
0:01:53.280,0:01:56.799
in the direction of the gradient
0:01:57.920,0:02:01.360
so classically if we have a screen over
0:02:00.079,0:02:02.719
here
0:02:01.360,0:02:04.399
some kind of measurement device for
0:02:02.719,0:02:05.840
measuring where the electrons land
0:02:04.399,0:02:09.840
classically we'd expect some kind of
0:02:05.840,0:02:09.840
spread that would look like this
0:02:11.039,0:02:14.080
and this would all be filled in
0:02:14.480,0:02:17.200
so it spreads out left to right just
0:02:15.840,0:02:19.680
because there's going to be some natural
0:02:17.200,0:02:23.200
spread of the beam
0:02:19.680,0:02:26.080
and in the top bottom direction
0:02:23.200,0:02:27.840
the electrons are being separated
0:02:26.080,0:02:30.239
according to the projection
0:02:27.840,0:02:31.760
of their angular momentum along the
0:02:30.239,0:02:34.640
field gradient direction
0:02:31.760,0:02:35.360
so if the if the electron happens to
0:02:34.640,0:02:36.879
be spinning
0:02:35.360,0:02:38.640
at 90 degrees to that field gradient
0:02:36.879,0:02:40.800
direction it has no magnetic field
0:02:38.640,0:02:42.160
in the direction of the gradient and so
0:02:40.800,0:02:43.200
it won't accelerate and we'll get it
0:02:42.160,0:02:45.280
into the middle
0:02:43.200,0:02:47.360
and on the other hand if it's completely
0:02:45.280,0:02:48.319
lined up with it either along or against
0:02:47.360,0:02:50.560
the gradient
0:02:48.319,0:02:53.599
it'll go to the top or bottom and we'd
0:02:50.560,0:02:54.800
expect it to take any value in between
0:02:53.599,0:02:56.800
but what they found when they did the
0:02:54.800,0:02:57.840
experiment is that it didn't look like
0:02:56.800,0:02:59.120
this at all
0:02:57.840,0:03:01.599
in fact the spread looks like something
0:02:59.120,0:03:04.800
like this that is
0:03:01.599,0:03:06.319
every electron either goes up or down
0:03:04.800,0:03:08.480
and there's nothing in between it's
0:03:06.319,0:03:10.480
quantized this is a very clear example
0:03:08.480,0:03:11.519
of the quantization of quantum mechanics
0:03:10.480,0:03:15.120
recalling that
0:03:11.519,0:03:16.640
quantum means discrete so it seems that
0:03:15.120,0:03:18.720
whenever we measure the spin of the
0:03:16.640,0:03:20.800
electron along any direction
0:03:18.720,0:03:22.480
it always takes one of two values and
0:03:20.800,0:03:25.760
those values are either
0:03:22.480,0:03:29.120
plus h bar over two or
0:03:25.760,0:03:31.760
minus h bar over two
0:03:29.120,0:03:33.920
so plus or minus a half in units of the
0:03:31.760,0:03:35.440
reduced planck's constant h bar
0:03:33.920,0:03:37.760
and so the electron is what we call
0:03:35.440,0:03:39.760
spin-half
0:03:37.760,0:03:40.959
so we can deduce various things from
0:03:39.760,0:03:43.760
applying these Stern-Gerlach
0:03:40.959,0:03:47.040
apparatuses to beams of electrons here
0:03:43.760,0:03:52.000
are the experimental observations
0:03:47.040,0:03:54.159
taking a Stern-Gerlach apparatus we can
0:03:52.000,0:03:55.760
place a block in front of one of the two
0:03:54.159,0:03:57.360
beams so for example we could block off
0:03:55.760,0:04:00.000
this beam down here
0:03:57.360,0:04:00.560
and then we'll be guaranteed to have
0:04:00.000,0:04:02.959
spin
0:04:00.560,0:04:04.640
plus a half or what's called spin up
0:04:02.959,0:04:06.159
in this direction and let's define this
0:04:04.640,0:04:08.879
direction to be z
0:04:06.159,0:04:10.080
in this case the Stern-Gerlach apparatus has a
0:04:08.879,0:04:11.519
direction in which it's going to split
0:04:10.080,0:04:13.200
the beam
0:04:11.519,0:04:15.120
and by blocking one of the paths we can
0:04:13.200,0:04:16.160
guarantee that our electrons are spin
0:04:15.120,0:04:17.840
polarized
0:04:16.160,0:04:18.880
in this case it would give us spin up we
0:04:17.840,0:04:20.560
could also choose the spin down
0:04:18.880,0:04:23.120
direction
0:04:20.560,0:04:24.160
so we make the following observation
0:04:23.120,0:04:26.000
measurement of spin
0:04:24.160,0:04:28.720
yields the values plus or minus h bar
0:04:26.000,0:04:30.479
over two only
0:04:28.720,0:04:32.160
consecutive measurements of the spin in
0:04:30.479,0:04:32.960
the same direction yield consistent
0:04:32.160,0:04:35.040
results
0:04:32.960,0:04:37.040
so if we pass this electron beam
0:04:35.040,0:04:37.919
selected out as plus h bar over two in
0:04:37.040,0:04:40.160
this direction
0:04:37.919,0:04:41.360
through a second Stern-Gerlach filter
0:04:40.160,0:04:43.199
in the same direction
0:04:41.360,0:04:44.160
we'll all the electrons that go
0:04:43.199,0:04:44.960
through the first will get through the
0:04:44.160,0:04:48.000
second
0:04:44.960,0:04:49.680
similarly if we put a second one in
0:04:48.000,0:04:51.919
the same direction and we block off all
0:04:49.680,0:04:53.520
the ones that have plus h bar over two
0:04:51.919,0:04:55.280
only allowing the ones with minus h bar
0:04:53.520,0:04:59.600
over two none will get through
0:04:55.280,0:05:02.479
because we'll get consistent results
0:04:59.600,0:05:04.000
however subsequent measurement in a
0:05:02.479,0:05:05.919
perpendicular direction
0:05:04.000,0:05:08.560
yields either the value plus or minus h
0:05:05.919,0:05:12.880
bar over two with equal probability
0:05:08.560,0:05:15.840
so that's the real clincher here
0:05:12.880,0:05:16.240
if we measure it in the spin in z and
0:05:15.840,0:05:18.639
then
0:05:16.240,0:05:20.240
we pass our spin polarized beam plus h
0:05:18.639,0:05:22.880
bar over two through a measurement in
0:05:20.240,0:05:25.280
x we'll have equal probability for it
0:05:22.880,0:05:27.360
to be plus a half or minus a half means
0:05:25.280,0:05:29.440
h bar but that's really weird because
0:05:27.360,0:05:32.479
that means if we perform a measurement
0:05:29.440,0:05:33.520
on in the z direction and then we
0:05:32.479,0:05:35.680
perform a measurement
0:05:33.520,0:05:38.960
in the x direction and then we form a
0:05:35.680,0:05:40.880
second measurement in the z direction
0:05:38.960,0:05:42.720
say we get plus h phi over 2 in the
0:05:40.880,0:05:44.960
first measurement of z
0:05:42.720,0:05:46.400
whatever we measure an x we select one
0:05:44.960,0:05:48.479
or the other of the two spin
0:05:46.400,0:05:50.000
polarizations in x and then we pass it
0:05:48.479,0:05:51.680
back to another z filter
0:05:50.000,0:05:53.680
well it said plus h bar over 2 the first
0:05:51.680,0:05:56.560
time but this time it has a 50:50
0:05:53.680,0:05:57.039
chance that any electrons coming
0:05:56.560,0:05:59.280
through this
0:05:57.039,0:06:00.639
to this have a 50:50 chance of going h bar
0:05:59.280,0:06:02.560
over 2 and 50:50
0:06:00.639,0:06:04.560
chance of giving the opposite so by
0:06:02.560,0:06:05.680
making an intermediate measurement in a
0:06:04.560,0:06:07.360
different direction
0:06:05.680,0:06:09.280
we can actually change the answer it
0:06:07.360,0:06:11.120
gives in the z direction and that's
0:06:09.280,0:06:12.639
one of the fundamental weird things
0:06:11.120,0:06:13.840
about quantum mechanics
0:06:12.639,0:06:15.120
so we're going to take a look at a
0:06:13.840,0:06:17.520
demonstration of that in a separate
0:06:15.120,0:06:17.520
video
0:06:17.840,0:06:22.319
for now let's just notice that there is
0:06:20.880,0:06:23.199
some precedent for this in classical
0:06:22.319,0:06:26.319
mechanics
0:06:23.199,0:06:28.000
if you think of let's take a spinning
0:06:26.319,0:06:30.240
classical object again
0:06:28.000,0:06:32.080
if we say this has some angular momentum
0:06:30.240,0:06:33.680
in this direction spinning like this
0:06:32.080,0:06:35.039
and we also want to say so this is a
0:06:33.680,0:06:36.400
some well defined value and i can work
0:06:35.039,0:06:37.919
out what that is
0:06:36.400,0:06:39.919
but if i also wanted to have a
0:06:37.919,0:06:40.639
well-defined angular momentum in this
0:06:39.919,0:06:42.000
direction
0:06:40.639,0:06:44.479
but it's got to spin like this and like
0:06:42.000,0:06:46.720
this so now what is its
0:06:44.479,0:06:48.560
spin in the first direction because when
0:06:46.720,0:06:49.840
it gets down to this it's now spinning
0:06:48.560,0:06:51.840
like this and so now it's got zero
0:06:49.840,0:06:53.840
angular momentum in this direction
0:06:51.840,0:06:54.960
but as it spins back round to here
0:06:53.840,0:06:57.520
then it's now got
0:06:54.960,0:06:58.720
the opposite of what it had before so
0:06:57.520,0:07:00.639
it's become time dependent
0:06:58.720,0:07:03.759
so there is some classical precedent
0:07:00.639,0:07:06.080
for this weirdness
0:07:03.759,0:07:07.120
but it is ultimately a fundamentally
0:07:06.080,0:07:17.840
quantum thing
0:07:07.120,0:07:17.840
spin and it's very weird
0:07:18.639,0:07:20.720
thank you
V5.3b Spin-1/2 (part II)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continued from video V5.3a) encoding the behaviours of spin-1/2 particles using a two-dimensional Hilbert space of vectors and matrices; probabilities and amplitudes for results of repeated measurements. Continued in V5.3c.
0:00:03.760,0:00:08.480
let's look at the mathematical structure
0:00:05.200,0:00:10.320
that we need to describe the situation
0:00:08.480,0:00:11.599
we're going to use matrices and
0:00:10.320,0:00:12.480
vectors for a reason we'll see in a
0:00:11.599,0:00:13.599
second
0:00:12.480,0:00:16.320
and so let's define the following
0:00:13.599,0:00:19.600
structure we'll define some
0:00:16.320,0:00:21.920
two by two matrices i
0:00:19.600,0:00:23.119
where i can be the direction x y or z
0:00:21.920,0:00:25.279
acting on a spin
0:00:23.119,0:00:27.199
up in that direction gives us the
0:00:25.279,0:00:29.760
value plus h bar over two
0:00:27.199,0:00:30.480
uh spin up so we want to find some
0:00:29.760,0:00:32.880
matrix
0:00:30.480,0:00:35.520
which has a normalized eigenvector whose
0:00:32.880,0:00:38.559
eigenvalue is plus h bar over two
0:00:35.520,0:00:41.600
and the other option
0:00:38.559,0:00:44.719
we'd like the same operator acting
0:00:41.600,0:00:45.360
on the other normalized eigenvector spin
0:00:44.719,0:00:47.200
down
0:00:45.360,0:00:49.360
gives us minus h bar over two because
0:00:47.200,0:00:52.160
these are the only two possible options
0:00:49.360,0:00:52.879
we'd like these matrices to be
0:00:52.160,0:00:55.039
hermitian
0:00:52.879,0:00:56.399
because we need the eigenvalues to be
0:00:55.039,0:00:58.879
these two values here
0:00:56.399,0:01:00.320
they need to be real because things we
0:00:58.879,0:01:03.280
measure in reality are real
0:01:00.320,0:01:04.080
rather than complex but also in
0:01:03.280,0:01:05.840
this case we know
0:01:04.080,0:01:07.360
the full set of eigenvalues they're
0:01:05.840,0:01:08.880
plus or minus h bar over
0:01:07.360,0:01:10.880
two since those are both real we must
0:01:08.880,0:01:13.920
have Hermitian matrix
0:01:10.880,0:01:15.600
okay why do we choose matrices well
0:01:13.920,0:01:18.960
we require
0:01:15.600,0:01:20.720
this to be the case we need objects
0:01:18.960,0:01:22.880
which tell us about the spins in the
0:01:20.720,0:01:24.640
different directions that do not commute
0:01:22.880,0:01:26.560
if they commuted remember if two
0:01:24.640,0:01:28.560
matrices commute we can find a
0:01:26.560,0:01:29.200
simultaneous set of eigenvectors for
0:01:28.560,0:01:30.960
them
0:01:29.200,0:01:32.640
but we can't have a simultaneous set
0:01:30.960,0:01:34.799
of eigenvectors for these two
0:01:32.640,0:01:36.400
operators because if we could then we
0:01:34.799,0:01:37.840
could say what the spin in the x
0:01:36.400,0:01:39.119
direction and what it is in the z
0:01:37.840,0:01:41.600
direction at the same time
0:01:39.119,0:01:43.119
we'd have eigenvalues of both would be
0:01:41.600,0:01:44.479
well defined at the same time
0:01:43.119,0:01:46.159
but we know this can't be the case
0:01:44.479,0:01:48.560
because we can do this z
0:01:46.159,0:01:49.759
x z measurement for example and the fact
0:01:48.560,0:01:52.000
that
0:01:49.759,0:01:53.759
so if we do repeated z in a row we'll
0:01:52.000,0:01:55.360
always get consistent results but if we
0:01:53.759,0:01:56.880
make an intermediate measurement
0:01:55.360,0:01:58.719
could get the opposite to result before
0:01:56.880,0:01:59.920
we got before so if you think about it
0:01:58.719,0:02:01.600
this structure tells you
0:01:59.920,0:02:02.960
that you must be using something which
0:02:01.600,0:02:03.920
has the properties of non-commuting
0:02:02.960,0:02:06.000
matrices
0:02:03.920,0:02:08.160
whether it's those or something weirder
0:02:06.000,0:02:10.479
is up to you but this is the
0:02:08.160,0:02:11.440
structure we've settled on and in fact
0:02:10.479,0:02:15.360
there's another condition
0:02:11.440,0:02:17.040
it's not just that you
0:02:15.360,0:02:18.480
get a random value in in the z
0:02:17.040,0:02:19.440
direction if you've measured in the x
0:02:18.480,0:02:21.440
direction before
0:02:19.440,0:02:22.879
your probability of plus or minus a half
0:02:21.440,0:02:24.319
is now fifty percent
0:02:22.879,0:02:25.760
so that additional structure actually
0:02:24.319,0:02:28.080
tells us that what we're looking for has
0:02:25.760,0:02:30.319
the following property
0:02:28.080,0:02:31.519
so the commutator of these two operators
0:02:30.319,0:02:34.480
remember the commutator
0:02:31.519,0:02:35.920
of matrices a and b is ab-ba
0:02:34.480,0:02:36.800
is how much they fail to commute with
0:02:35.920,0:02:39.040
each other
0:02:36.800,0:02:39.920
the commutator of two of these spin
0:02:39.040,0:02:42.000
operators
0:02:39.920,0:02:44.000
should be i h bar times the other spin
0:02:42.000,0:02:46.560
operator and sign here will depend on
0:02:44.000,0:02:48.000
the order in which you do these
0:02:46.560,0:02:49.680
okay so we'll take a look at how
0:02:48.000,0:02:51.120
this exact structure comes about
0:02:49.680,0:02:54.400
in the problem sets
0:02:51.120,0:02:56.000
for now we can identify a set of
0:02:54.400,0:02:59.040
three two by two matrices that have
0:02:56.000,0:03:01.040
these properties and they're as follows
0:02:59.040,0:03:02.159
so the h bar over two times these
0:03:01.040,0:03:03.680
matrices sigma i
0:03:02.159,0:03:06.640
where these are the Pauli matrices
0:03:03.680,0:03:06.640
defined as follows
0:03:06.720,0:03:10.319
so the three matrices defined like this
0:03:08.879,0:03:11.840
they're not the only matrices that will
0:03:10.319,0:03:13.760
have these properties but they're a
0:03:11.840,0:03:17.040
particularly convenient choice
0:03:13.760,0:03:20.159
for what we'd like to do so
0:03:17.040,0:03:22.879
in particular let's take sigma z here
0:03:20.159,0:03:24.720
it's just one zero zero minus one and
0:03:22.879,0:03:26.560
we can see that the normalized
0:03:24.720,0:03:28.480
eigenvectors for that matrix must be
0:03:26.560,0:03:31.920
this
0:03:28.480,0:03:32.480
so multiply sigma z by h bar over 2
0:03:31.920,0:03:35.360
to get...
0:03:32.480,0:03:36.879
the spin operator in the z
0:03:35.360,0:03:39.200
direction
0:03:36.879,0:03:41.200
and that is just h bar over 2 times that
0:03:39.200,0:03:44.080
sigma z so acting on this vector 1
0:03:41.200,0:03:45.760
0 must return h bar over 2 1 0. so
0:03:44.080,0:03:48.640
that's good that's got the structure
0:03:45.760,0:03:49.360
of our spin up and as said acting on
0:03:48.640,0:03:51.360
zero one
0:03:49.360,0:03:53.200
is minus h bar over two zero one so
0:03:51.360,0:03:55.280
that's the structure of our spin down
0:03:53.200,0:03:56.400
and you can check that your two
0:03:55.280,0:03:58.959
eigenvectors
0:03:56.400,0:04:01.120
for both of these matrices have the
0:03:58.959,0:04:04.799
same properties
0:04:01.120,0:04:08.239
okay so let's just copy our
0:04:04.799,0:04:11.920
z eigenvectors again and
0:04:08.239,0:04:14.480
the spin up value spin up in x
0:04:11.920,0:04:15.760
eigenvector is as follows
0:04:14.480,0:04:17.919
which you can readily check from the
0:04:15.760,0:04:21.120
matrix itself and
0:04:17.919,0:04:25.280
we can decompose this into spin up and
0:04:21.120,0:04:25.280
spin down in z directions as follows
0:04:25.440,0:04:29.360
because we can add one zero and zero one
0:04:27.280,0:04:30.560
to get one one and multiply by the three
0:04:29.360,0:04:33.680
factor pre-factor to
0:04:30.560,0:04:36.080
normalize and we get r up in x
0:04:33.680,0:04:37.759
eigenvector this i didn't have to take
0:04:36.080,0:04:39.040
this form but we knew that we must be
0:04:37.759,0:04:42.400
able to write
0:04:39.040,0:04:45.680
we can write any two pi 2 complex
0:04:42.400,0:04:48.720
vector in terms of our z
0:04:45.680,0:04:51.520
eigen vectors because
0:04:48.720,0:04:52.560
these are the eigenvectors of a
0:04:51.520,0:04:55.199
non-degenerate
0:04:52.560,0:04:56.160
Hermitian matrix and remember from a
0:04:55.199,0:04:58.720
previous video
0:04:56.160,0:05:00.160
that the full set of normalized
0:04:58.720,0:05:01.280
eigenvectors of a non-degenerate
0:05:00.160,0:05:03.440
Hermitian matrix
0:05:01.280,0:05:04.560
must form a complete orthonormal basis
0:05:03.440,0:05:07.360
so we can expand
0:05:04.560,0:05:08.560
any vector in that space in terms of a
0:05:07.360,0:05:10.639
complete set of these and there's only
0:05:08.560,0:05:12.880
two of them in this case
0:05:10.639,0:05:14.960
okay so why do we want to do this well
0:05:12.880,0:05:18.000
say we've prepared a state
0:05:14.960,0:05:19.840
in say up in
0:05:18.000,0:05:21.360
x we pass it through a Stern-Gerlach
0:05:19.840,0:05:24.080
apparatus in the x direction
0:05:21.360,0:05:26.160
we select the beam that's spin polarized
0:05:24.080,0:05:29.280
plus a half
0:05:26.160,0:05:32.400
in units of h bar but now we want to
0:05:29.280,0:05:33.280
pass it through a z apparatus and look
0:05:32.400,0:05:34.639
at the results
0:05:33.280,0:05:37.680
of what's coming out so we want to
0:05:34.639,0:05:40.720
schematically want to do this
0:05:37.680,0:05:42.639
so this is schematic for the
0:05:40.720,0:05:44.800
polarisation in the
0:05:42.639,0:05:47.600
x direction
0:05:44.800,0:05:49.840
we pass some beam of electrons
0:05:47.600,0:05:52.639
into here we select out only the plus
0:05:49.840,0:05:53.360
h bar of the two results the others we
0:05:52.639,0:05:55.520
throw away
0:05:53.360,0:05:57.120
into the screen so everything now coming
0:05:55.520,0:05:59.199
over here and the beam is
0:05:57.120,0:06:00.479
getting redirected through some
0:05:59.199,0:06:01.600
magic or you can
0:06:01.600,0:06:05.120
use electric fields to
0:06:03.039,0:06:06.319
redirect it doesn't have to be magic
0:06:05.120,0:06:08.479
so we're going to take this set of
0:06:06.319,0:06:09.280
electrons where their spin is known to
0:06:08.479,0:06:11.840
be
0:06:09.280,0:06:13.919
plus h bar over two in the x direction
0:06:11.840,0:06:15.120
and we pass it into the apparatus
0:06:13.919,0:06:17.199
in the z direction
0:06:15.120,0:06:18.479
and we want to know what are the
0:06:17.199,0:06:22.800
probabilities for getting
0:06:18.479,0:06:26.000
h bar over 2 minus h bar over 2.
0:06:22.800,0:06:26.880
well we take our result here up in x we
0:06:26.000,0:06:29.360
decompose it
0:06:26.880,0:06:30.240
into the z basis so we we have these
0:06:29.360,0:06:33.520
results
0:06:30.240,0:06:34.240
and then the amplitude for finding up in
0:06:33.520,0:06:36.880
z
0:06:34.240,0:06:38.160
is just given by the coefficient of
0:06:36.880,0:06:42.240
up in z
0:06:38.160,0:06:44.960
okay so we have this result
0:06:42.240,0:06:47.360
the amplitude for measuring up in z for
0:06:44.960,0:06:50.080
a state which is known to be up in x
0:06:47.360,0:06:51.599
is given by the inner product up in z
0:06:50.080,0:06:54.479
acting on in our x
0:06:51.599,0:06:54.960
that's right acting on upper up in x
0:06:54.479,0:06:57.280
okay
0:06:54.960,0:06:59.039
so we start from our initial state and
0:06:57.280,0:06:59.440
we act what we want our final state to
0:06:59.039,0:07:01.280
be
0:06:59.440,0:07:02.960
and that'll give us the amplitude if we
0:07:01.280,0:07:06.479
do that up here we find
0:07:02.960,0:07:09.680
we're going to act up instead like this
0:07:06.479,0:07:11.120
from the left it comes through let's not
0:07:09.680,0:07:13.360
worry about this one
0:07:11.120,0:07:16.000
acted on here we get up in z up in z
0:07:13.360,0:07:18.639
let's rewrite it
0:07:16.000,0:07:20.240
so we get the inner product of up in
0:07:18.639,0:07:22.880
z without being said but we know that
0:07:20.240,0:07:26.479
must be one
0:07:22.880,0:07:28.720
up in z with down and z is zero
0:07:26.479,0:07:30.160
because again the basis is orthogonal
0:07:28.720,0:07:34.240
and so we just get the answer
0:07:30.160,0:07:35.919
1 over root 2. you could have done it by
0:07:34.240,0:07:39.520
writing it out in this particular choice
0:07:35.919,0:07:41.440
of basis but you didn't need to
0:07:39.520,0:07:43.039
so that's the amplitude but that
0:07:41.440,0:07:44.639
amplitude is not what gives
0:07:43.039,0:07:46.080
us the probabilities:
0:07:44.639,0:07:47.840
use the Born rule it's the modulus
0:07:46.080,0:07:50.240
square of the amplitude and so we have
0:07:47.840,0:07:52.319
the result
0:07:50.240,0:07:53.599
the modulus square of the previous
0:07:52.319,0:07:54.800
result
0:07:53.599,0:07:56.800
this is actually a totally general
0:07:54.800,0:07:58.560
statement that works for
0:07:56.800,0:08:00.240
not just spin half not just finite
0:07:58.560,0:08:01.680
dimensional spaces throughout all of
0:08:00.240,0:08:03.440
quantum mechanics we can say the
0:08:01.680,0:08:05.520
following
0:08:03.440,0:08:07.199
the amplitude for measuring a state psi_final
0:08:05.520,0:08:10.319
given a state
0:08:07.199,0:08:11.120
prepared as phi_initial is given by the
0:08:10.319,0:08:15.120
inner product
0:08:11.120,0:08:18.720
of psi_final with phi_initial
0:08:15.120,0:08:20.560
and the corresponding probability
0:08:18.720,0:08:22.960
is given as the Born rule tells us by
0:08:20.560,0:08:25.039
the modulus square
0:08:22.960,0:08:26.000
which can be written like this where phi
0:08:25.039,0:08:29.919
i have just abbreviated
0:08:26.000,0:08:31.520
phi_initial by
0:08:29.919,0:08:33.279
i know it's very exciting isn't it Geoffrey
0:08:31.520,0:08:37.200
it's completely general doesn't just
0:08:33.279,0:08:39.120
apply to this particular case okay
0:08:37.200,0:08:40.719
okay so we'd like to perform repeated
0:08:39.120,0:08:41.440
measurements of the following
0:08:40.719,0:08:43.519
form
0:08:41.440,0:08:45.360
we'll send in a beam we don't need to
0:08:43.519,0:08:50.240
know what it is
0:08:45.360,0:08:50.240
we'll pass it through a z-oriented apparatus
0:08:50.320,0:08:55.839
we'll take only the plus h bar over 2
0:08:53.279,0:08:55.839
results
0:08:55.920,0:09:01.600
feed it into a y apparatus
0:09:03.040,0:09:09.839
we'll take let's say the down results
0:09:10.320,0:09:17.839
put that into an x
0:09:13.600,0:09:17.839
and we'll take the up
0:09:19.200,0:09:22.959
this is all totally arbitrary i'm just
0:09:21.279,0:09:24.560
making sure i'm sticking to
0:09:22.959,0:09:28.160
the notes that you're
0:09:24.560,0:09:28.160
getting handed to you
0:09:29.040,0:09:32.800
and finally we'll put it back into
0:09:30.320,0:09:35.200
another z but let's see the probability
0:09:32.800,0:09:37.839
for it to come out
0:09:35.200,0:09:37.839
down
0:09:40.720,0:09:44.080
how do we do this okay well let's start
0:09:43.279,0:09:45.839
at this point
0:09:44.080,0:09:47.360
this is the first time we know the
0:09:45.839,0:09:51.839
state of the particle
0:09:47.360,0:09:51.839
and we know here that it's up in z
0:09:52.480,0:09:55.519
so what's the state here clearly it must
0:09:54.800,0:09:58.640
be state
0:09:55.519,0:09:58.640
down in y
0:10:01.120,0:10:04.560
right because it's just come out of the
0:10:02.399,0:10:08.240
y Stern-Gerlach apparatus
0:10:04.560,0:10:11.600
what's the amplitude for it to
0:10:08.240,0:10:13.839
be in that state well it was up in z
0:10:11.600,0:10:14.640
so we want the amplitude for it to be
0:10:13.839,0:10:17.680
down in y
0:10:14.640,0:10:19.760
given that it was up in z
0:10:17.680,0:10:21.120
and we know that from the previous board
0:10:19.760,0:10:24.880
that's given by
0:10:21.120,0:10:27.360
down in y given it was initially up
0:10:24.880,0:10:27.360
in z
0:10:29.680,0:10:33.040
okay so it's got in this state remember
0:10:31.680,0:10:33.920
this is an inner product so this is just
0:10:33.040,0:10:36.320
a complex
0:10:33.920,0:10:37.200
number so this is our amplitude so it's
0:10:36.320,0:10:40.160
in this state
0:10:37.200,0:10:41.360
this is the amplitude because it was
0:10:40.160,0:10:44.399
initially in this state
0:10:41.360,0:10:45.279
and it must finally be in this state we
0:10:44.399,0:10:48.560
can rewrite this
0:10:45.279,0:10:50.720
slightly more neatly as follows
0:10:48.560,0:10:52.160
so again this is the state this is just
0:10:50.720,0:10:53.519
an amplitude but i pulled it over to the
0:10:52.160,0:10:56.560
other side so i get this nice
0:10:53.519,0:10:59.120
outer product here okay
0:10:56.560,0:11:00.160
over here it's going to be in the
0:10:59.120,0:11:03.760
state
0:11:00.160,0:11:04.959
up in x but what's the amplitude for it
0:11:03.760,0:11:08.560
being in that state
0:11:04.959,0:11:10.640
well this was the in-going state
0:11:08.560,0:11:12.079
and that's the outgoing state so we know
0:11:10.640,0:11:16.320
that our amplitude
0:11:12.079,0:11:16.320
is going to be as follows
0:11:16.560,0:11:20.560
so it's going to be complicated but all
0:11:18.640,0:11:21.200
it is is this previous state here this
0:11:20.560,0:11:22.640
is the
0:11:21.200,0:11:24.640
initial state as far as this is
0:11:22.640,0:11:27.440
concerned the final state
0:11:24.640,0:11:28.399
is up in x so we take inner product of
0:11:27.440,0:11:30.079
final state
0:11:28.399,0:11:31.920
with initial state and that's our
0:11:30.079,0:11:33.200
amplitude and we multiply it by the
0:11:31.920,0:11:36.959
state itself
0:11:33.200,0:11:38.800
so rewriting this slightly we have
0:11:36.959,0:11:40.800
so i've just pulled the state through
0:11:38.800,0:11:42.160
the amplitude again just as before
0:11:40.800,0:11:44.320
you actually see what we're doing every
0:11:42.160,0:11:45.519
time it's just multiplying from the left
0:11:44.320,0:11:49.120
by the outer product
0:11:45.519,0:11:50.720
of the relative measurement in each case
0:11:49.120,0:11:53.440
and so finally we'd like to say what's
0:11:50.720,0:11:55.519
the amplitude for it to end up down in z
0:11:53.440,0:11:57.519
well that means that we're going to
0:11:55.519,0:12:01.200
create the state down in z so let's
0:11:57.519,0:12:01.200
act from the left which is down in z
0:12:05.279,0:12:13.839
onto this so it's up in x
0:12:11.200,0:12:13.839
down in y
0:12:16.560,0:12:20.320
and it started off as up and down
0:12:21.680,0:12:25.360
so the amplitude for it to end up in
0:12:23.519,0:12:28.399
this state here
0:12:25.360,0:12:28.399
is just given by this
0:12:31.279,0:12:35.600
let's rewrite that so it's a chain of
0:12:34.720,0:12:38.560
conditional
0:12:35.600,0:12:39.360
statements the amplitude for down in z
0:12:38.560,0:12:41.120
at the end
0:12:39.360,0:12:43.519
but it was this is conditional on the
0:12:41.120,0:12:45.120
fact that it was up in x previously
0:12:43.519,0:12:47.040
that's conditional on it being down in y
0:12:45.120,0:12:49.040
before that and that's conditional on it
0:12:47.040,0:12:50.560
being initially up in z
0:12:49.040,0:12:52.079
it's given by this amplitude here which
0:12:50.560,0:12:53.839
is just the amplitude of the state we
0:12:52.079,0:12:56.560
just evaluated
0:12:53.839,0:12:59.040
and we happen to know that each of these
0:12:56.560,0:13:00.560
different things you can work it out
0:12:59.040,0:13:02.959
using the matrices or we can just use
0:13:00.560,0:13:04.320
the fact that the probability for it
0:13:02.959,0:13:05.760
being
0:13:04.320,0:13:08.480
measured in any particular given state
0:13:05.760,0:13:10.320
in z given a certain state in x
0:13:08.480,0:13:11.519
and the same is true for any
0:13:10.320,0:13:13.440
orthogonal measurement
0:13:11.519,0:13:15.600
the amplitude must be one over root
0:13:13.440,0:13:18.959
two so the probability is a half
0:13:15.600,0:13:20.720
so we get the following one over root
0:13:18.959,0:13:23.360
two cubed
0:13:20.720,0:13:26.800
and so the probability is one over two
0:13:23.360,0:13:26.800
cubed which is one over eight
0:13:26.959,0:13:29.600
so we're going to use this kind of chain
0:13:28.399,0:13:32.000
of reasoning when we take a look at the
0:13:29.600,0:13:35.839
demonstration of this type of effect
0:13:32.000,0:13:35.839
in a separate video
V5.3c Spin-1/2 (part III)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continued from video V5.3b) using the mathematical structure introduced to model spin-1/2 particles in order to derive the resolution of the identity; expectation values of the spin operator.
0:00:03.840,0:00:07.200
okay we can use this
0:00:05.040,0:00:10.559
structure to prove a mathematical result
0:00:07.200,0:00:13.920
that we've deduced mathematically before
0:00:10.559,0:00:14.960
so say we send some state it could be
0:00:13.920,0:00:17.600
unknown it can be known it doesn't
0:00:14.960,0:00:19.199
matter into a Stern-Gerlach apparatus
0:00:17.600,0:00:20.960
and let's say it's in the z direction
0:00:19.199,0:00:25.199
also doesn't matter
0:00:20.960,0:00:28.320
so let's call this state psi unknown
0:00:25.199,0:00:32.880
we split it into its two parts
0:00:28.320,0:00:32.880
but we're going to recombine them
0:00:33.520,0:00:36.719
now we haven't actually done anything
0:00:35.600,0:00:37.920
right because we haven't blocked either
0:00:36.719,0:00:40.079
of these beams off
0:00:37.920,0:00:41.600
we separate the spin polarizations but
0:00:40.079,0:00:44.480
then we recombine them
0:00:41.600,0:00:45.760
now remember that it's amplitudes
0:00:44.480,0:00:47.120
which are relevant to quantum mechanics
0:00:45.760,0:00:49.120
rather than probabilities
0:00:47.120,0:00:50.960
we can add the amplitudes back
0:00:49.120,0:00:52.719
together and we get the original state
0:00:50.960,0:00:54.160
so this just gives us our original state
0:00:52.719,0:00:55.520
back
0:00:54.160,0:00:57.120
we haven't performed a measurement
0:00:55.520,0:00:58.000
before when we block one of the beams
0:00:57.120,0:00:59.520
we're effectively performing
0:00:58.000,0:01:02.160
a measurement by saying if it's gone
0:00:59.520,0:01:04.000
past here we know that it must be
0:01:02.160,0:01:05.760
spin up instead we're not going to do
0:01:04.000,0:01:09.200
that now so
0:01:05.760,0:01:12.400
in the intermediate state we have
0:01:09.200,0:01:15.840
up in z here
0:01:12.400,0:01:18.479
and the amplitude for that state must be
0:01:15.840,0:01:20.159
it was initially in psi and it was
0:01:18.479,0:01:24.840
finally in up in z
0:01:20.159,0:01:28.320
so it was initial psi
0:01:24.840,0:01:29.920
final up in z
0:01:28.320,0:01:32.960
and similarly down here this would be in
0:01:29.920,0:01:32.960
state down in z
0:01:33.600,0:01:40.479
initially psi final state was
0:01:37.119,0:01:42.960
down in z
0:01:40.479,0:01:43.600
and so we find that since this psi
0:01:42.960,0:01:46.960
equals this
0:01:43.600,0:01:50.159
psi we must have that
0:01:46.960,0:01:51.680
the state psi this one over here
0:01:50.159,0:01:54.960
has to be the initial psi which is the
0:01:51.680,0:01:57.920
same the amplitude of that to be up in z
0:01:54.960,0:01:59.360
multiplied by the state up in z plus the
0:01:57.920,0:02:01.200
amplitude for it to be down on z
0:01:59.360,0:02:02.560
multiplied by the state down in z
0:02:01.200,0:02:04.479
because
0:02:02.560,0:02:05.920
it's either this or this for classical
0:02:04.479,0:02:08.319
probabilities we would
0:02:05.920,0:02:09.599
add the two options here it's
0:02:08.319,0:02:11.680
amplitude switch add rather than
0:02:09.599,0:02:13.680
probabilities so we get this
0:02:11.680,0:02:15.440
and i've rewritten the amplitudes to
0:02:13.680,0:02:18.080
the right of the states
0:02:15.440,0:02:19.680
but now i can factor out i can see
0:02:18.080,0:02:22.959
that everything multiplies a psi
0:02:19.680,0:02:25.920
on the right so i can rewrite this
0:02:22.959,0:02:26.720
so psi is equal to up in z outer product
0:02:25.920,0:02:29.360
up in z
0:02:26.720,0:02:30.000
plus down in z outer product down in z
0:02:30.000,0:02:33.599
and this whole thing acting on acting on
0:02:31.680,0:02:35.280
the psi
0:02:33.599,0:02:36.800
and since this is true for any psi it
0:02:35.280,0:02:39.120
doesn't matter which
0:02:36.800,0:02:39.840
it must be the case that this object
0:02:39.120,0:02:41.840
here
0:02:39.840,0:02:44.480
is just doing the job of the identity
0:02:41.840,0:02:47.680
matrix and so we have the final result
0:02:44.480,0:02:49.680
the identity matrix is equal to
0:02:47.680,0:02:51.280
up-in-z outer product up-in-z plus
0:02:49.680,0:02:53.280
down-in-z outer product down-in-z
0:02:51.280,0:02:54.480
you can check it explicitly in the basis
0:02:53.280,0:02:57.280
where up in z is one
0:02:54.480,0:02:58.480
zero down in z is zero one but
0:02:57.280,0:03:01.120
actually you can deduce it
0:02:58.480,0:03:04.080
just using the results of these
0:03:01.120,0:03:05.599
experiments which is very nice
0:03:04.080,0:03:08.000
and so this is the resolution of the
0:03:05.599,0:03:11.040
identity which we deduced
0:03:08.000,0:03:11.920
mathematically before and here we're
0:03:11.040,0:03:13.440
just
0:03:11.920,0:03:15.200
relying ultimately on the fact that this
0:03:13.440,0:03:19.040
structure has
0:03:15.200,0:03:19.599
our observables the the values of the
0:03:19.040,0:03:22.239
spin
0:03:19.599,0:03:22.879
as the eigenvalues of hermitian matrices
0:03:22.879,0:03:26.480
non-degenerate Hermitian matrices
0:03:25.200,0:03:27.760
and those non-degenerate Hermitian
0:03:26.480,0:03:29.760
matrices must have
0:03:27.760,0:03:31.200
complete orthonormal bases of
0:03:29.760,0:03:33.920
eigenvectors
0:03:31.200,0:03:35.519
explicitly you can write it as follows
0:03:33.920,0:03:36.640
using this particular choice of basis and
0:03:35.519,0:03:39.519
you can check it yourself
0:03:36.640,0:03:40.799
and you can also check it works for the
0:03:39.519,0:03:42.480
spins in the x direction and the spins
0:03:40.799,0:03:46.400
in the y direction
0:03:42.480,0:03:47.840
okay earlier on in the course when we're
0:03:46.400,0:03:48.720
dealing with wave functions which we
0:03:47.840,0:03:51.760
will return to
0:03:48.720,0:03:53.200
later we looked at expectation values
0:03:51.760,0:03:54.239
of things like the position and the
0:03:53.200,0:03:56.400
momentum
0:03:54.239,0:03:57.840
we can look at expectation values of
0:03:56.400,0:04:01.200
spin operators as well
0:03:57.840,0:04:03.040
and we get something like this that is
0:04:01.200,0:04:05.519
the expectation value
0:04:03.040,0:04:07.439
of the operator this is the spin
0:04:05.519,0:04:10.400
in the i direction x y or z
0:04:07.439,0:04:11.840
evaluated for state psi we can just
0:04:10.400,0:04:15.280
write this in direct notation
0:04:11.840,0:04:18.079
as the matrix S_i acting on
0:04:15.280,0:04:19.040
ket psi sandwiched with bra psi so
0:04:18.079,0:04:20.239
it's like a bracket
0:04:19.040,0:04:21.759
with the operator sandwiched in the
0:04:20.239,0:04:23.040
middle and so it's another convenience
0:04:21.759,0:04:25.919
of direct notation
0:04:23.040,0:04:26.800
the expectation value we use in
0:04:25.919,0:04:28.639
statistics
0:04:26.800,0:04:30.320
you know subject separated from quantum
0:04:28.639,0:04:31.759
mechanics we would expect to write the
0:04:30.320,0:04:32.639
expectation value with these angled
0:04:31.759,0:04:34.639
brackets
0:04:32.639,0:04:36.720
and so by construction now the
0:04:34.639,0:04:38.400
mathematical operation we do to find it
0:04:36.720,0:04:40.400
is built into the notation so that's
0:04:38.400,0:04:42.960
very convenient so in particular we
0:04:40.400,0:04:46.880
could evaluate the following
0:04:42.960,0:04:49.600
the expectation value of the
0:04:46.880,0:04:52.000
S_z operator the spin in the z direction
0:04:49.600,0:04:55.600
for an eigenstate of the S_z operator
0:04:52.000,0:04:57.680
is just given by the operator
0:04:55.600,0:05:00.400
acting on up in z just gives h bar over
0:04:57.680,0:05:02.240
two acting up in z by definition
0:05:00.400,0:05:05.600
that's how we constructed the
0:05:02.240,0:05:08.240
mathematics and so this evaluates to
0:05:05.600,0:05:10.240
h bar over two the eigenvalue which
0:05:08.240,0:05:12.800
hopefully makes sense
0:05:10.240,0:05:14.320
if we were look look instead of this the
0:05:12.800,0:05:17.360
expectation value of the
0:05:14.320,0:05:18.080
spin x operator for a state which is up
0:05:17.360,0:05:21.280
in z
0:05:18.080,0:05:23.120
we know we must get the result zero
0:05:21.280,0:05:26.639
because there's an equal probability
0:05:23.120,0:05:27.120
of up in zed giving the value up in x as
0:05:26.639,0:05:29.680
there is
0:05:27.120,0:05:30.720
down in x it must always give one of
0:05:29.680,0:05:32.639
those results because that's the
0:05:30.720,0:05:33.600
measurement that's being performed
0:05:32.639,0:05:35.120
and you can check this structure
0:05:33.600,0:05:36.160
mathematically and we'll do that in the
0:05:35.120,0:05:39.840
problems
0:05:36.160,0:05:39.840
okay thank you for your time
V5.4 Polarisation demo
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
a demonstration showing some interesting properties of the polarisation of light, used to confirm the relevance of the mathematical structure employing non-commuting matrices employed in V5.3 for spin-1/2 systems.
0:00:00.640,0:00:05.279
hello in a previous video we saw
0:00:03.040,0:00:06.319
some mathematical properties of spin
0:00:05.279,0:00:07.759
half systems
0:00:06.319,0:00:09.760
so we were thinking about electrons with
0:00:07.759,0:00:11.519
their spins we were passing them
0:00:09.760,0:00:13.120
through stern gerlach apparatuses which
0:00:11.519,0:00:14.799
separate them according to whether they
0:00:13.120,0:00:17.680
have spin up or spin down
0:00:14.799,0:00:20.160
in a given direction and we saw various
0:00:17.680,0:00:21.279
interesting properties that you can find
0:00:20.160,0:00:22.720
so i don't have a Stern Gerlach
0:00:21.279,0:00:24.400
apparatus to show you in my bedroom
0:00:22.720,0:00:27.519
unfortunately
0:00:24.400,0:00:29.279
but i do have an equivalent experiment
0:00:27.519,0:00:30.240
we can do involving the polarization of
0:00:29.279,0:00:32.000
light
0:00:30.240,0:00:33.920
so in the stern gerlach case with the
0:00:32.000,0:00:36.000
spin of the electron
0:00:33.920,0:00:38.480
the orthogonal states and the quantum
0:00:36.000,0:00:39.760
mechanical sense are spin up and spin
0:00:38.480,0:00:42.160
down
0:00:39.760,0:00:43.120
in polarization its vertical
0:00:42.160,0:00:46.079
polarization
0:00:43.120,0:00:47.520
and horizontal polarization so remember
0:00:46.079,0:00:48.320
the polarization of light you can think
0:00:47.520,0:00:49.840
of as
0:00:48.320,0:00:51.600
the direction which the electric field
0:00:49.840,0:00:53.280
is oscillating
0:00:51.600,0:00:55.199
if we take a polarization filter like
0:00:53.280,0:00:56.719
this you can see that it's blocking some
0:00:55.199,0:00:58.000
of the light
0:00:56.719,0:01:00.320
so it should be blocking about half the
0:00:58.000,0:01:00.879
light and what it's doing is letting in
0:01:00.320,0:01:03.039
the light
0:01:00.879,0:01:04.239
that has a linear polarization so an
0:01:03.039,0:01:04.879
electric field oscillating in one
0:01:04.239,0:01:07.840
direction
0:01:04.879,0:01:09.040
and let's say it's up in this case
0:01:07.840,0:01:10.400
so one of the things we saw in the
0:01:09.040,0:01:13.360
stern-gerlach case
0:01:10.400,0:01:14.720
was that if you can prepare an
0:01:13.360,0:01:17.600
electron which certainly has
0:01:14.720,0:01:18.720
a spin up in a given direction say z
0:01:17.600,0:01:22.080
and then you perform
0:01:18.720,0:01:23.119
another measurement on spin up in the z
0:01:22.080,0:01:25.280
direction that's all right
0:01:23.119,0:01:27.040
another measurement in the z direction
0:01:25.280,0:01:28.080
it's guaranteed to give the answer
0:01:27.040,0:01:31.200
spin up again
0:01:28.080,0:01:31.840
okay so in the stern gerlach case we we
0:01:31.200,0:01:33.840
split
0:01:31.840,0:01:35.680
the a beam of electrons and spin up
0:01:33.840,0:01:37.439
and spin down then we block the down
0:01:35.680,0:01:38.240
beam and now this remaining beam is all
0:01:37.439,0:01:39.600
spin up
0:01:38.240,0:01:41.520
we pass it through a second Stern Gerlach
0:01:39.600,0:01:44.720
apparatus in the same direction
0:01:41.520,0:01:46.560
and we again block the down beam
0:01:44.720,0:01:47.520
and we find that everything we got
0:01:46.560,0:01:48.479
through the first one got through the
0:01:47.520,0:01:50.720
second one
0:01:48.479,0:01:52.159
and similarly if we measure in the same
0:01:50.720,0:01:53.759
direction the second time but we tried
0:01:52.159,0:01:54.000
to let only down through on the second
0:01:53.759,0:01:55.600
one
0:01:54.000,0:01:57.840
and up through on the first one then
0:01:55.600,0:01:58.799
zero percent of them will get through
0:01:57.840,0:02:01.280
so we can see all of that with
0:01:58.799,0:02:02.799
polarization filters so here's
0:02:01.280,0:02:04.799
a polarization filter i'll bring it
0:02:02.799,0:02:06.960
quite close so you can see it okay
0:02:04.799,0:02:08.640
so you're looking at polarized light
0:02:06.960,0:02:09.440
through there now i'll bring a second
0:02:08.640,0:02:10.720
one in
0:02:09.440,0:02:12.319
and you see there should be basically no
0:02:10.720,0:02:14.080
change because all the light that goes
0:02:12.319,0:02:15.520
through one gets through the other
0:02:14.080,0:02:17.200
okay so now i'm going to take a second
0:02:15.520,0:02:18.959
i'm going to rotate it 90 degrees which
0:02:17.200,0:02:21.040
is the equivalent of spin up to spin
0:02:18.959,0:02:22.319
down
0:02:21.040,0:02:24.480
and you see that now no light gets
0:02:22.319,0:02:26.160
through okay because
0:02:24.480,0:02:28.000
if it's getting through the first one it
0:02:26.160,0:02:29.040
must be vertically polarized if it's
0:02:28.000,0:02:30.640
getting through the second one it must
0:02:29.040,0:02:32.959
be horizontally polarized
0:02:30.640,0:02:34.080
and nothing can have both of those
0:02:32.959,0:02:36.959
it's effectively
0:02:34.080,0:02:38.480
making a spin half measurement in the
0:02:36.959,0:02:41.200
same direction twice
0:02:38.480,0:02:42.800
and asking us two opposite things but
0:02:41.200,0:02:45.840
the really weird thing we saw with
0:02:42.800,0:02:47.680
spin was that if we measure in z
0:02:45.840,0:02:48.879
say and we find it to be spin up
0:02:47.680,0:02:50.160
if we perform another measurement
0:02:48.879,0:02:51.519
instead we'll find it to be spin up
0:02:50.160,0:02:52.480
again and we can do measure it as many
0:02:51.519,0:02:53.760
times as we like
0:02:52.480,0:02:55.440
but if we perform an intermediate
0:02:53.760,0:02:56.720
measurement in a different direction 90
0:02:55.440,0:02:59.440
degrees say spin
0:02:56.720,0:03:00.959
in the x direction it'll have a 50
0:02:59.440,0:03:02.480
chance of being either up or down in the
0:03:00.959,0:03:03.840
x direction
0:03:02.480,0:03:06.560
but then it's well defined in the x
0:03:03.840,0:03:07.599
direction we select out say all the ups
0:03:06.560,0:03:10.000
in the x direction
0:03:07.599,0:03:11.120
and now we pass it to a second z filter
0:03:10.000,0:03:12.560
and now it has a 50:50
0:03:11.120,0:03:14.879
chance anything that gets through that x
0:03:12.560,0:03:16.319
filter has a 50:50 chance of being either
0:03:14.879,0:03:18.720
up or down in z
0:03:16.319,0:03:20.319
so whereas if we keep measuring instead
0:03:18.720,0:03:22.000
we'll keep finding the same answer if we
0:03:20.319,0:03:23.360
make an intermediate measurement in 90
0:03:22.000,0:03:26.080
degree rotated direction
0:03:23.360,0:03:26.640
we will randomize the result so we can
0:03:26.080,0:03:29.760
do that with
0:03:26.640,0:03:30.799
polarization filters so again here's one
0:03:29.760,0:03:32.560
filter
0:03:30.799,0:03:33.760
here's a second filter there's a 90
0:03:32.560,0:03:34.720
degrees let's just check here's
0:03:33.760,0:03:38.000
everything getting through
0:03:34.720,0:03:40.000
there's nothing getting through and now
0:03:38.000,0:03:42.480
if i bring a third filter
0:03:40.000,0:03:44.400
if i put it in front of the first two
0:03:42.480,0:03:46.560
nothing will happen
0:03:44.400,0:03:47.599
if i put it behind the first two nothing
0:03:46.560,0:03:49.760
will happen
0:03:47.599,0:03:50.879
and if i put it between the two
0:03:49.760,0:03:53.040
something will happen
0:03:50.879,0:03:56.159
there you go you can see me let's try
0:03:53.040,0:03:56.159
that at that angle
0:03:57.680,0:04:01.280
there we go so you can see me through
0:04:00.000,0:04:02.959
that right
0:04:01.280,0:04:04.480
so here nothing getting through i put an
0:04:02.959,0:04:07.040
additional filter in
0:04:04.480,0:04:08.000
and something gets through so what we're
0:04:07.040,0:04:11.360
doing is
0:04:08.000,0:04:11.920
we're measuring we were only
0:04:11.360,0:04:13.200
letting
0:04:11.920,0:04:15.920
vertically polarized things through this
0:04:13.200,0:04:17.199
one if we let only
0:04:15.920,0:04:18.799
horizontally polarized through this one
0:04:17.199,0:04:20.079
nothing gets through but if in the
0:04:18.799,0:04:20.880
meantime we make an intermediate
0:04:20.079,0:04:24.000
measurement
0:04:20.880,0:04:25.520
at 45 degrees like this which is the
0:04:24.000,0:04:28.400
equivalent of a spin x measurement on
0:04:25.520,0:04:30.000
something that was spin z before
0:04:28.400,0:04:31.759
and then we pass our second filter now
0:04:30.000,0:04:33.280
something gets through
0:04:31.759,0:04:35.840
okay so it's a little demonstration you
0:04:33.280,0:04:37.600
can do at home which demonstrates
0:04:35.840,0:04:38.160
these mathematical properties of spin
0:04:37.600,0:04:40.960
half
0:04:38.160,0:04:42.400
in general it's a two-dimensional
0:04:40.960,0:04:44.400
hilbert space
0:04:42.400,0:04:48.880
of quantum mechanical particles all
0:04:44.400,0:04:48.880
right thank you for your time
V6.1 Operators and observables
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
matrix mechanics: representing observable quantities by Hermitian operators (matrices or differential operators).
0:00:00.080,0:00:04.400
hello in this video we're going to take
0:00:02.000,0:00:07.520
a look at operators and observables
0:00:04.400,0:00:09.280
in particular we have the following
0:00:07.520,0:00:11.360
observable quantities which we usually
0:00:09.280,0:00:13.599
abbreviate to observables
0:00:11.360,0:00:15.200
are represented by Hermitian operators in
0:00:13.599,0:00:17.520
quantum mechanics
0:00:15.200,0:00:18.880
so let's look at an example first and it
0:00:17.520,0:00:20.880
should be very familiar to us at this
0:00:18.880,0:00:23.680
point
0:00:20.880,0:00:24.240
it's just the hamiltonian which you may
0:00:23.680,0:00:26.720
recall
0:00:24.240,0:00:27.599
is referred to as the energy operator
0:00:26.720,0:00:30.640
and it forms
0:00:30.640,0:00:34.079
the time independent schrodinger
0:00:32.719,0:00:37.280
equation here
0:00:34.079,0:00:39.840
where these states or kets phi
0:00:37.280,0:00:40.960
subscript n are the eigen states of the
0:00:39.840,0:00:42.879
hamiltonian
0:00:40.960,0:00:44.960
and E_n are corresponding
0:00:42.879,0:00:47.600
eigen energies
0:00:44.960,0:00:48.559
so we've seen this written in a
0:00:47.600,0:00:50.399
differential
0:00:48.559,0:00:52.960
form as a differential operator which
0:00:50.399,0:00:55.039
looks like this
0:00:52.960,0:00:57.199
so we've written it in three dimensions
0:00:55.039,0:01:00.800
as minus h bar squared over 2m
0:00:57.199,0:01:04.720
grad squared plus v acting on
0:01:00.800,0:01:05.119
phi_n(x) and V we've occasionally written
0:01:04.720,0:01:07.280
as V hat
0:01:05.119,0:01:09.200
here indicating it's an operator
0:01:07.280,0:01:12.000
when it acts on a function of position
0:01:09.200,0:01:13.119
like this it simply becomes which
0:01:12.000,0:01:16.560
hopefully you can read
0:01:13.119,0:01:18.479
is V and then in parentheses x so
0:01:16.560,0:01:19.920
V(x) the potential
0:01:18.479,0:01:22.080
this term here which may have looked a
0:01:19.920,0:01:24.159
little bit mysterious
0:01:22.080,0:01:25.920
since we know that the hamiltonian
0:01:24.159,0:01:27.200
corresponds to an energy and an energy
0:01:25.920,0:01:28.479
should have a kinetic part and a
0:01:27.200,0:01:30.720
potential part
0:01:28.479,0:01:31.680
and V(x) is the potential part we can
0:01:30.720,0:01:34.799
deduce that this
0:01:31.680,0:01:36.320
should be the kinetic part
0:01:34.799,0:01:38.400
the kinetic energy should be p
0:01:36.320,0:01:39.200
squared over 2m so this leads us to
0:01:38.400,0:01:41.280
deduce
0:01:39.200,0:01:43.920
that this quantity should probably be
0:01:41.280,0:01:43.920
written as
0:01:44.079,0:01:50.159
some operator p so p hat squared
0:01:47.360,0:01:51.920
over 2m and this p is what we call the
0:01:50.159,0:01:53.680
momentum operator
0:01:51.920,0:01:55.759
so we've mainly worked with the
0:01:53.680,0:01:59.040
hamiltonian
0:01:55.759,0:02:01.759
it can be a finite dimensional
0:01:59.040,0:02:03.520
matrix big n by big n and we've seen an
0:02:01.759,0:02:05.600
example of a finite dimensional
0:02:03.520,0:02:07.119
complex Hilbert space in the previous
0:02:05.600,0:02:08.319
video where we're looking at
0:02:07.119,0:02:10.479
spin one-half
0:02:08.319,0:02:13.280
so in finite dimensional systems the
0:02:10.479,0:02:16.720
hamiltonian will be a matrix
0:02:13.280,0:02:18.400
in fact the phrase 'operator' refers to
0:02:16.720,0:02:20.720
matrices in finite numbers of
0:02:18.400,0:02:22.959
dimensions but it also applies to
0:02:20.720,0:02:24.560
infinite numbers of dimensions and when
0:02:22.959,0:02:27.200
we have an infinite dimensional
0:02:24.560,0:02:28.480
hilbert space what we're really
0:02:27.200,0:02:31.280
talking about is
0:02:28.480,0:02:32.640
the matrix hamiltonian turns into this
0:02:31.280,0:02:33.920
differential operator
0:02:32.640,0:02:35.840
so we've actually already done the hard
0:02:33.920,0:02:37.280
case the infinite dimensional space that
0:02:35.840,0:02:39.519
sounds daunting but actually
0:02:37.280,0:02:41.599
all it means is that we're working for
0:02:39.519,0:02:44.080
example in the position basis
0:02:41.599,0:02:45.680
like this and so our states are labeled
0:02:44.080,0:02:47.760
by their functions
0:02:45.680,0:02:49.360
and they're labeled by a position but a
0:02:47.760,0:02:50.080
position can take any of an infinite
0:02:49.360,0:02:52.319
number of values
0:02:50.080,0:02:53.680
along the line it's a real number and
0:02:52.319,0:02:56.959
for each of those we would like to think
0:02:53.680,0:02:58.640
of it as a different basis vector
0:02:56.959,0:03:00.319
so what we're saying in terms of
0:02:58.640,0:03:02.560
matrices and vectors is something like
0:03:00.319,0:03:02.560
this
0:03:02.640,0:03:06.159
so it's a bit strange it looks like a
0:03:04.239,0:03:10.000
tautology but we're saying there exists
0:03:06.159,0:03:10.640
a position operator x hat
0:03:10.640,0:03:13.920
and there are eigen states of this
0:03:12.159,0:03:14.400
operator and we'll denote those with a
0:03:13.920,0:03:17.360
ket
0:03:14.400,0:03:19.840
with an x in it and the eigenvalues of
0:03:17.360,0:03:23.040
the x operator acting on ket x
0:03:19.840,0:03:24.720
comes out as the position x the
0:03:23.040,0:03:27.760
real number
0:03:24.720,0:03:29.680
okay so we know such
0:03:27.760,0:03:30.959
an operator must exist so we posit it to
0:03:29.680,0:03:32.720
exist in our structure
0:03:30.959,0:03:34.480
because positions are observable
0:03:32.720,0:03:35.120
quantities we can see a particle at a
0:03:34.480,0:03:36.640
position
0:03:35.120,0:03:38.879
so that this is a quantity we can assign
0:03:36.640,0:03:40.239
to it
0:03:38.879,0:03:42.560
but also i'm saying that we should have
0:03:40.239,0:03:45.760
this other operator p and so we deduce
0:03:42.560,0:03:47.840
that must obey the following equation
0:03:45.760,0:03:48.959
it looks similarly tautological but bear
0:03:47.840,0:03:51.040
in mind again that these are different
0:03:48.959,0:03:53.680
things this is an operator
0:03:51.040,0:03:55.840
either a matrix or a differential
0:03:53.680,0:03:57.760
operator
0:03:55.840,0:03:59.200
this is an eigen state of that operator
0:03:57.760,0:04:01.599
and this is an eigenvalue which is just
0:03:59.200,0:04:04.799
a real number
0:04:01.599,0:04:08.000
so as we'll see
0:04:04.799,0:04:09.760
in a later video the momentum
0:04:08.000,0:04:11.840
so the position and momentum operators
0:04:09.760,0:04:15.120
when written in real space
0:04:11.840,0:04:16.720
are as follows okay so working in what
0:04:15.120,0:04:17.680
we call the position basis in quantum
0:04:16.720,0:04:19.759
mechanics writing
0:04:17.680,0:04:20.880
things as functions
0:04:19.759,0:04:24.160
of position
0:04:20.880,0:04:27.199
the x the position operator is simply
0:04:24.160,0:04:29.440
x the position
0:04:27.199,0:04:30.720
in real space and the momentum operator
0:04:29.440,0:04:32.320
p
0:04:30.720,0:04:34.560
we can deduce from this that p squared
0:04:32.320,0:04:35.840
over 2m is equal to minus h bar squared
0:04:34.560,0:04:38.880
grad squared over 2m
0:04:35.840,0:04:39.440
but up to a sign p the operator has to
0:04:38.880,0:04:42.720
be minus
0:04:39.440,0:04:45.919
i h bar grad or in one dimension minus i
0:04:42.720,0:04:47.120
h bar d/dx so we'll take a look at
0:04:45.919,0:04:48.400
what that really means in a couple of
0:04:47.120,0:04:50.960
videos' time
0:04:48.400,0:04:52.320
for now let's work entirely with the
0:04:50.960,0:04:55.120
operators
0:04:52.320,0:04:56.479
so when heisenberg wrote down what's
0:04:55.120,0:04:59.360
called matrix mechanics
0:04:56.479,0:05:00.479
which is what we're really studying here
0:05:00.479,0:05:05.840
he chose to use matrices and their
0:05:04.160,0:05:07.360
infinite dimensional generalizations
0:05:05.840,0:05:09.600
which are operators
0:05:07.360,0:05:11.039
for the following reason that matrices
0:05:09.600,0:05:14.000
need not commute
0:05:11.039,0:05:14.960
and in fact he took from experiments
0:05:14.000,0:05:17.840
the following
0:05:14.960,0:05:20.160
kind of mathematically intuitive
0:05:17.840,0:05:20.160
guess
0:05:20.400,0:05:24.000
so he guessed that the position operator
0:05:22.400,0:05:25.280
and the momentum operator
0:05:24.000,0:05:27.360
he knew that they shouldn't
0:05:25.280,0:05:28.880
commute
0:05:27.360,0:05:30.720
the reason they shouldn't commute
0:05:28.880,0:05:31.120
is much like we saw in the previous
0:05:30.720,0:05:32.880
video
0:05:31.120,0:05:34.479
we looked at spin one-half systems
0:05:32.880,0:05:35.840
remember in that case the Stern Gerlach
0:05:34.479,0:05:38.639
experiment tells us
0:05:35.840,0:05:39.199
that if we have definite information
0:05:38.639,0:05:42.000
about
0:05:39.199,0:05:44.160
the spin in the x direction we must be
0:05:42.000,0:05:47.600
completely uncertain about its value in
0:05:44.160,0:05:50.479
the y and z directions and so
0:05:47.600,0:05:51.520
this suggests a structure like matrices
0:05:50.479,0:05:54.400
because
0:05:51.520,0:05:55.280
if two matrices commute then they can
0:05:54.400,0:05:58.319
have a simultaneous
0:05:55.280,0:06:00.000
set of eigenvectors
0:05:58.319,0:06:01.360
but if they don't commute then they
0:06:00.000,0:06:02.639
can't have a simultaneous set of
0:06:01.360,0:06:04.960
eigenvectors
0:06:02.639,0:06:06.080
and so it admits a structure in which we
0:06:04.960,0:06:08.960
can have things like
0:06:06.080,0:06:09.840
operator x and operator p and if they
0:06:08.960,0:06:11.759
don't commute
0:06:09.840,0:06:13.360
means that we can't know x and p at the
0:06:11.759,0:06:17.120
same time
0:06:13.360,0:06:18.479
so Heisenberg took this from
0:06:17.120,0:06:20.160
the various pieces of experimental
0:06:18.479,0:06:23.199
information and guessed
0:06:20.160,0:06:24.400
that we can try and phrase the behaviour
0:06:23.199,0:06:26.160
of quantum particles in terms of
0:06:24.400,0:06:28.479
operators like this
0:06:26.160,0:06:30.400
and in particular the inspired guess
0:06:28.479,0:06:31.840
of his is what's called the 'canonical
0:06:30.400,0:06:34.240
commutation relation' which is what's
0:06:31.840,0:06:36.479
written here
0:06:34.240,0:06:37.919
which is the commutator of x and p how
0:06:36.479,0:06:41.440
much they fail to commute
0:06:37.919,0:06:42.960
is given by i h bar and this fat one
0:06:41.440,0:06:45.840
with a hat on it is called
0:06:42.960,0:06:47.919
the identity operator and it's a
0:06:45.840,0:06:49.440
trivial operator which acts on any state
0:06:47.919,0:06:51.440
gives the state back so it's like the
0:06:49.440,0:06:54.080
'one' of operators
0:06:51.440,0:06:55.039
so it's got the reduced planck's
0:06:54.080,0:06:56.479
constant in it
0:06:55.039,0:06:57.840
which is good because generally that
0:06:56.479,0:06:59.599
should appear if we're studying quantum
0:06:57.840,0:07:02.639
problems
0:06:59.599,0:07:05.280
and it's got an i in it which
0:07:02.639,0:07:07.039
is so there's a mathematical reason that
0:07:05.280,0:07:07.520
this must appear but you can see that in
0:07:07.039,0:07:10.000
general
0:07:07.520,0:07:11.280
complex numbers are important to
0:07:10.000,0:07:14.080
quantum mechanics
0:07:11.280,0:07:15.360
so let's put a box around this so this
0:07:14.080,0:07:16.160
is a really important relation which
0:07:15.360,0:07:19.280
we'll come back to
0:07:16.160,0:07:22.400
repeatedly in this course so
0:07:19.280,0:07:24.479
we'd like to try and represent
0:07:22.400,0:07:26.319
observable quantities with things like
0:07:24.479,0:07:27.919
matrices because then we can have the
0:07:26.319,0:07:29.440
property that they don't commute and we
0:07:27.919,0:07:30.720
don't necessarily we may not be able to
0:07:29.440,0:07:31.759
have simultaneous knowledge of two
0:07:30.720,0:07:33.039
different ones
0:07:31.759,0:07:35.440
the reason they should be Hermitian
0:07:33.039,0:07:37.440
is that Hermitian matrices have
0:07:35.440,0:07:39.599
real eigenvalues and the general
0:07:37.440,0:07:41.520
Hermitian operators have real eigenvalues
0:07:39.599,0:07:43.759
and every number we measure in
0:07:41.520,0:07:46.160
reality is a real number
0:07:43.759,0:07:48.160
so that's why we have this structure in
0:07:46.160,0:07:50.400
quantum mechanics
0:07:48.160,0:07:53.680
okay so let's put this to some use to
0:07:50.400,0:07:57.199
derive a very famous result
0:07:53.680,0:08:00.160
the heisenberg uncertainty principle
0:07:57.199,0:08:02.240
so we define the uncertainty in a
0:08:00.160,0:08:06.160
quantity to be the standard deviation
0:08:02.240,0:08:07.039
as follows so the standard deviation of
0:08:06.160,0:08:10.000
a quantity
0:08:07.039,0:08:10.639
is given by the square of the
0:08:10.000,0:08:13.039
quantity
0:08:10.639,0:08:14.240
and the average over that minus the
0:08:13.039,0:08:17.440
average of the quantity
0:08:14.240,0:08:20.000
squared then we square root that and
0:08:17.440,0:08:21.039
there's a not too difficult but somewhat
0:08:20.000,0:08:23.919
lengthy
0:08:21.039,0:08:25.919
derivation of the following result the
0:08:23.919,0:08:27.520
product of the uncertainties
0:08:25.919,0:08:29.199
in operators A and B and their
0:08:27.520,0:08:31.840
corresponding observables
0:08:29.199,0:08:33.599
must be greater than or equal to half
0:08:31.840,0:08:36.560
multiplied by the modulus
0:08:33.599,0:08:38.240
of the average of the commutator of the
0:08:36.560,0:08:41.279
quantities
0:08:38.240,0:08:43.519
or the corresponding operators
0:08:41.279,0:08:45.440
so in particular we could take the
0:08:43.519,0:08:47.040
case of the canonical commutation
0:08:45.440,0:08:50.959
relations
0:08:47.040,0:08:53.040
substituted in and we find that the
0:08:50.959,0:08:54.880
product of the uncertainties of position
0:08:53.040,0:08:55.440
and momentum must be greater than or
0:08:54.880,0:08:58.480
equal to
0:08:55.440,0:08:58.480
h bar over two
0:09:00.720,0:09:03.839
so the uncertainty principle tells us
0:09:02.560,0:09:05.200
that the more accurately we know the
0:09:03.839,0:09:07.200
position of the particle
0:09:05.200,0:09:09.680
the less accurately we're able to know
0:09:07.200,0:09:11.360
the momentum and vice versa
0:09:09.680,0:09:13.040
now it's an inherently quantum
0:09:11.360,0:09:14.880
mechanical effect but there is a
0:09:13.040,0:09:18.240
classical precedent for it
0:09:14.880,0:09:21.120
if we take a rope and we establish a
0:09:18.240,0:09:22.720
standing wave along that rope
0:09:21.120,0:09:26.160
so let me try and get you one there we
0:09:22.720,0:09:26.160
go one with a node in the center
0:09:28.160,0:09:34.160
there we go so
0:09:31.279,0:09:35.680
looking at that standing wave
0:09:34.160,0:09:36.880
it's possible to say what the wavelength
0:09:35.680,0:09:38.160
is
0:09:36.880,0:09:40.000
you can see that there was a full
0:09:38.160,0:09:41.600
wavelength along the rope's length there
0:09:40.000,0:09:43.600
or in this case half a wavelength
0:09:41.600,0:09:46.000
along the rope's length
0:09:43.600,0:09:47.839
I can know the wavelength perfectly
0:09:46.000,0:09:49.839
and then from the de Broglie relation we
0:09:47.839,0:09:52.160
can assign a momentum to that
0:09:49.839,0:09:54.320
wave but then if you ask the question
0:09:52.160,0:09:56.640
where is the particle or
0:09:54.320,0:09:59.040
where is the wave located clearly it's
0:09:56.640,0:10:01.920
along the entire length of the rope
0:09:59.040,0:10:02.160
so it's as unknown as it's possible to
0:10:01.920,0:10:03.360
be
0:10:02.160,0:10:05.200
it's completely spread out along the
0:10:03.360,0:10:08.240
rope. On the other hand
0:10:05.200,0:10:11.040
if I take the rope and I whip it
0:10:08.240,0:10:13.440
I can send a disturbance on the rope
0:10:11.040,0:10:15.680
like that
0:10:13.440,0:10:16.800
and then you can say fairly accurately
0:10:15.680,0:10:18.640
where the
0:10:16.800,0:10:20.800
disturbance is and it's acting kind of
0:10:18.640,0:10:23.040
like a particle
0:10:20.800,0:10:24.800
but if you're to ask what the wavelength
0:10:23.040,0:10:27.760
of that
0:10:24.800,0:10:28.800
disturbance is it doesn't have one right
0:10:27.760,0:10:30.800
it's not got
0:10:28.800,0:10:32.000
the same shape as a wave in fact what
0:10:30.800,0:10:34.640
you'd have to do is carry out
0:10:32.000,0:10:36.880
Fourier analysis and you'd find that
0:10:34.640,0:10:40.079
that disturbance is described by
0:10:36.880,0:10:41.920
an infinite set of the possible
0:10:40.079,0:10:44.160
standing waves on the rope
0:10:41.920,0:10:45.600
so you can know the position of the
0:10:44.160,0:10:46.240
disturbance or you can know the
0:10:45.600,0:10:49.519
wavelength
0:10:46.240,0:10:50.959
and from that deduce the momentum
0:10:49.519,0:10:52.480
you can't know both simultaneously and
0:10:50.959,0:10:55.040
the more you know one the less you know
0:10:52.480,0:10:57.200
the other and vice versa
0:10:55.040,0:10:58.880
so there are several different
0:10:57.200,0:11:01.440
uncertainty principles
0:10:58.880,0:11:02.880
or different sets of pairs of operators
0:11:01.440,0:11:04.320
and observables
0:11:02.880,0:11:08.160
which have uncertainty relations with
0:11:04.320,0:11:11.440
them so one is position momentum
0:11:08.160,0:11:12.880
another is energy and time we have to be
0:11:11.440,0:11:13.440
a bit careful about what we mean by this
0:11:12.880,0:11:15.440
one
0:11:13.440,0:11:17.200
and it's difficult to quantify it in
0:11:15.440,0:11:18.720
quite the same sense mathematically
0:11:17.200,0:11:20.720
because there's no time operator in
0:11:18.720,0:11:22.399
quantum mechanics time is special: things
0:11:20.720,0:11:25.839
are just a function of time
0:11:22.399,0:11:26.160
but we know that there is some kind
0:11:25.839,0:11:27.760
of an
0:11:26.160,0:11:29.680
uncertainty relation between these two
0:11:27.760,0:11:31.600
because for example if we have a
0:11:29.680,0:11:33.360
particle which is going to decay after
0:11:31.600,0:11:36.560
some finite amount of time
0:11:33.360,0:11:38.800
the shorter lived the particle
0:11:36.560,0:11:39.839
(so the more accurately we know
0:11:38.800,0:11:41.839
the decay time)
0:11:39.839,0:11:43.360
the less accurately we know its energy
0:11:41.839,0:11:44.959
and vice versa
0:11:43.360,0:11:46.800
and there are various other energy time
0:11:44.959,0:11:48.640
uncertainties we can form
0:11:46.800,0:11:50.240
again you can sort of get a classical
0:11:48.640,0:11:51.760
idea of this by bearing in mind that the
0:11:50.240,0:11:54.160
energy is related to the
0:11:51.760,0:11:55.760
frequency of the particle and in order
0:11:54.160,0:11:56.240
to establish the frequency of some kind
0:11:56.240,0:12:00.079
of periodically changing wave you need
0:11:58.880,0:12:01.600
to measure a few
0:12:00.079,0:12:03.120
cycles of it in order to get the
0:12:01.600,0:12:04.959
frequency
0:12:03.120,0:12:06.560
so if you localize it too much in time
0:12:04.959,0:12:09.200
you can't get the frequency information
0:12:06.560,0:12:10.399
so you lose the energy information
0:12:09.200,0:12:12.240
we've seen there's an uncertainty
0:12:10.399,0:12:14.720
relation between
0:12:12.240,0:12:16.560
spin in different directions we if we
0:12:14.720,0:12:18.240
know the spin in the x direction say
0:12:16.560,0:12:20.800
we have complete uncertainty about the
0:12:18.240,0:12:22.959
spin in the y direction
0:12:20.800,0:12:24.399
spin is a form of angular momentum the
0:12:22.959,0:12:25.920
intrinsic angular momentum
0:12:24.399,0:12:29.600
and in fact more generally angular
0:12:25.920,0:12:31.360
momentum as we'll see in future videos
0:12:29.600,0:12:32.720
also is an observable in quantum
0:12:31.360,0:12:34.480
mechanics and there is also
0:12:32.720,0:12:37.600
an uncertainty relation between angular
0:12:34.480,0:12:39.600
momentum in different directions
0:12:37.600,0:12:42.079
and there's also an uncertainty relation
0:12:39.600,0:12:43.680
for superconductors and superfluids
0:12:42.079,0:12:45.920
which aren't part of this course
0:12:43.680,0:12:47.279
but in that case you have a condensate
0:12:45.920,0:12:48.560
with many different particles in it
0:12:47.279,0:12:51.120
forming a macroscopic
0:12:48.560,0:12:52.560
quantum mechanical wave function and the
0:12:51.120,0:12:54.480
more accurately you know the number of
0:12:52.560,0:12:56.000
particles in the condensate
0:12:54.480,0:12:59.279
the less accurately you can know the
0:12:56.000,0:13:00.959
phase of the condensate vice versa
0:12:59.279,0:13:02.480
so this is a list of some of the
0:13:00.959,0:13:03.519
uncertainty relations which come up in
0:13:02.480,0:13:05.279
quantum mechanics
0:13:03.519,0:13:07.120
you'll notice the top two actually are
0:13:05.279,0:13:09.920
fourier transforms of one another
0:13:07.120,0:13:11.120
so another classical precedent for the
0:13:09.920,0:13:14.240
uncertainty relation
0:13:11.120,0:13:17.200
is in signals analysis
0:13:14.240,0:13:18.720
where if you localize some kind of
0:13:17.200,0:13:21.279
signal in
0:13:18.720,0:13:23.279
the direct space the fourier transform
0:13:21.279,0:13:23.920
signal is less well localized and vice
0:13:23.279,0:13:25.360
versa
0:13:23.920,0:13:27.279
you can think of this in terms of a
0:13:25.360,0:13:28.560
gaussian for example if you Fourier
0:13:27.279,0:13:29.440
transform a gaussian you get another
0:13:28.560,0:13:30.720
gaussian back
0:13:29.440,0:13:32.800
but if you make the direct space
0:13:30.720,0:13:35.519
gaussian narrower so you know
0:13:32.800,0:13:41.519
the location of it more accurately the
0:13:35.519,0:13:41.519
fourier transform gets broader.
V6.2 The Heisenberg picture
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
two ways of understanding the time evolution of quantum systems: the Schrödinger picture (time-dependent states and time-independent operators) and the Heisenberg picture (time-independent states and time-dependent operators); the independence of expectation values on picture; the Heisenberg equation of motion. Continued in video V6.3
0:00:00.080,0:00:05.120
hello in this video we're going to take
0:00:02.560,0:00:07.919
a look at the heisenberg picture
0:00:05.120,0:00:08.880
so in the first part of this course we
0:00:07.919,0:00:10.160
looked at
0:00:08.880,0:00:12.639
what's really called the Schroedinger
0:00:10.160,0:00:15.040
picture that is we were describing
0:00:12.639,0:00:16.240
the system in terms of wave functions
0:00:15.040,0:00:18.240
and these wave functions
0:00:16.240,0:00:20.320
had a time dependence to them and they
0:00:18.240,0:00:22.080
were acted on by operators
0:00:20.320,0:00:25.439
such as the hamiltonian which were
0:00:22.080,0:00:25.439
themselves time independent
0:00:25.519,0:00:28.880
so in the schrodinger picture we had
0:00:27.119,0:00:32.320
time dependent states
0:00:28.880,0:00:34.399
acted on by time independent operators
0:00:32.320,0:00:35.440
our states we wrote as the wave function
0:00:34.399,0:00:37.520
psi(x,t)
0:00:35.440,0:00:38.559
I'm just writing it here as a
0:00:37.520,0:00:40.879
ket
0:00:38.559,0:00:42.840
psi which is still a function of time in
0:00:40.879,0:00:44.719
the heisenberg picture we have the
0:00:42.840,0:00:47.760
following
0:00:44.719,0:00:51.120
time independent states acted on by time
0:00:47.760,0:00:54.480
dependent operators so
0:00:51.120,0:00:56.079
there are two fundamentally different
0:00:54.480,0:00:57.680
ways of looking quantum mechanics which
0:00:56.079,0:00:58.960
we're come up with by schrodinger and
0:00:57.680,0:01:01.840
heisenberg
0:00:58.960,0:01:03.199
and others at the same time so
0:01:01.840,0:01:06.159
schrodinger's picture
0:01:03.199,0:01:08.479
has this format and he was thinking of
0:01:06.159,0:01:11.040
everything in terms of
0:01:08.479,0:01:12.400
an extension of classical wave theory
0:01:11.040,0:01:13.040
whereas heisenberg was thinking of
0:01:12.400,0:01:15.840
things
0:01:15.840,0:01:19.680
in terms of
0:01:16.880,0:01:21.119
non-commuting matrices and their
0:01:19.680,0:01:22.320
infinite dimensional counterparts as
0:01:21.119,0:01:25.360
operators
0:01:22.320,0:01:26.159
so it was later that year shown by
0:01:25.360,0:01:28.320
schrodinger
0:01:26.159,0:01:30.320
that the two pictures are completely
0:01:28.320,0:01:33.680
equivalent
0:01:30.320,0:01:34.000
so the mathematics of one or the other
0:01:33.680,0:01:35.520
may
0:01:34.000,0:01:37.119
make certain problems easier to deal
0:01:35.520,0:01:37.680
with we'll look at some problems in this
0:01:37.119,0:01:39.040
video
0:01:37.680,0:01:41.119
where the heisenberg picture is the more
0:01:39.040,0:01:43.200
natural setting
0:01:41.119,0:01:44.399
but the two can always be interchanged
0:01:43.200,0:01:47.200
and in fact what you have
0:01:44.399,0:01:49.680
is you have your complex hilbert space
0:01:47.200,0:01:52.159
it has states living in it
0:01:49.680,0:01:53.360
and you want to carry out procedures
0:01:52.159,0:01:54.479
which are going to transform you from
0:01:53.360,0:01:57.840
one state to another
0:01:54.479,0:01:59.200
just like acting matrices on vectors to
0:01:57.840,0:02:01.040
switch them to other vectors that live
0:01:59.200,0:02:03.119
in the same vector space
0:02:01.040,0:02:05.360
so the question is really just one of
0:02:03.119,0:02:07.360
active versus passive transformations
0:02:05.360,0:02:09.039
just like you can either rotate your
0:02:07.360,0:02:10.239
vector and keep your coordinates fixed
0:02:09.039,0:02:11.599
or you can keep your vector fixed and
0:02:10.239,0:02:13.520
rotate your coordinates
0:02:11.599,0:02:14.959
that's really the fundamental
0:02:13.520,0:02:16.239
difference that's happening here so it's
0:02:14.959,0:02:19.040
not a physical difference it's a
0:02:16.239,0:02:19.040
mathematical one
0:02:19.599,0:02:23.599
so within the schrodinger picture we
0:02:21.920,0:02:26.400
have time dependent states
0:02:23.599,0:02:27.440
so let's label them psi_s
0:02:26.400,0:02:29.680
for schrodinger
0:02:27.440,0:02:31.440
as a function of time but we know that
0:02:29.680,0:02:33.360
psi_s(t)
0:02:31.440,0:02:34.800
is nothing other than psi_s at time
0:02:33.360,0:02:38.800
equals zero
0:02:34.800,0:02:42.080
acted on by this unitary operator
0:02:38.800,0:02:44.879
e to the minus i hamiltonian
0:02:42.080,0:02:46.239
times time divided by h bar so we've
0:02:44.879,0:02:48.480
seen this in the previous video
0:02:46.239,0:02:49.920
but really you can just think
0:02:48.480,0:02:50.640
back to the time dependent Schroedinger
0:02:49.920,0:02:52.560
equation
0:02:50.640,0:02:54.800
and remember you can separate that
0:02:52.560,0:02:56.160
equation -- it's a separable equation
0:02:54.800,0:02:58.239
the time dependent part really just
0:02:56.160,0:02:58.560
tells us this in general we would have
0:02:58.239,0:03:00.560
this
0:02:58.560,0:03:02.159
H sorry in the specific case we looked
0:03:00.560,0:03:04.800
at before we'd have H
0:03:02.159,0:03:05.440
here would be the energy eigenvalue
0:03:04.800,0:03:07.360
solving
0:03:05.440,0:03:08.640
the time dependent schrodinger equation
0:03:07.360,0:03:10.000
but totally generally
0:03:08.640,0:03:11.920
if we don't want to work with energy
0:03:10.000,0:03:13.200
eigenstates just arbitrary states of the
0:03:11.920,0:03:15.519
hilbert space
0:03:13.200,0:03:17.440
this is the hamiltonian itself so this
0:03:15.519,0:03:18.319
is an exponential of a hermitian
0:03:17.440,0:03:21.120
operator
0:03:18.319,0:03:22.480
but it's still just an operator you can
0:03:21.120,0:03:25.280
think of exponentials
0:03:22.480,0:03:27.280
of operators or exponentials of matrices
0:03:25.280,0:03:29.280
defined in terms of their taylor series
0:03:27.280,0:03:31.040
so it's an infinite power series in
0:03:29.280,0:03:32.879
terms of the hamiltonian
0:03:31.040,0:03:34.879
and then this quantity is what's called
0:03:32.879,0:03:37.599
unitary which you may recall
0:03:34.879,0:03:39.680
means that preserves kets' inner
0:03:37.599,0:03:42.480
products
0:03:39.680,0:03:43.680
so a convenient choice to take to relate
0:03:42.480,0:03:46.319
to the schrodinger picture to the
0:03:43.680,0:03:49.280
heisenberg is as follows
0:03:46.319,0:03:50.159
let's define our time independent
0:03:49.280,0:03:52.720
heisenberg
0:03:50.159,0:03:53.680
ket our state psi subscript H for
0:03:52.720,0:03:55.680
Heisenberg
0:03:53.680,0:03:56.720
in the hilbert space let's take it by
0:03:55.680,0:03:59.519
definition to be
0:03:56.720,0:04:01.360
the schrodinger state at time equals
0:03:59.519,0:04:04.319
zero
0:04:01.360,0:04:04.720
and then other choices are completely
0:04:04.319,0:04:07.200
fine
0:04:04.720,0:04:07.760
if we were to pick some random time 10
0:04:07.200,0:04:10.080
seconds
0:04:07.760,0:04:11.760
instead of zero here all it would do is
0:04:10.080,0:04:14.959
multiply this by a
0:04:11.760,0:04:17.199
complex phase of unit magnitude because
0:04:14.959,0:04:20.000
of this
0:04:17.199,0:04:21.919
but if you multiply the state by some
0:04:20.000,0:04:23.360
unit magnitude complex phase
0:04:21.919,0:04:25.280
well that's just the global phase of the
0:04:23.360,0:04:25.840
wave function; the wave function in this
0:04:25.280,0:04:28.320
case
0:04:25.840,0:04:29.919
being the the ket or the state here and
0:04:28.320,0:04:32.800
if you multiply the state
0:04:29.919,0:04:34.160
by a global phase it doesn't change
0:04:32.800,0:04:37.199
anything physically as we know
0:04:34.160,0:04:39.360
global phases are unobservable so
0:04:37.199,0:04:41.360
let's choose this one by definition it's
0:04:39.360,0:04:44.240
just a convenient choice
0:04:41.360,0:04:45.759
this gives us the following relation
0:04:44.240,0:04:48.479
just combining the
0:04:45.759,0:04:49.199
equations we have that the time
0:04:48.479,0:04:51.600
dependent
0:04:49.199,0:04:53.040
state in the schrodinger picture is just
0:04:51.600,0:04:56.160
given by e to the minus
0:04:53.040,0:04:56.880
i hamiltonian times time divided by h
0:04:56.160,0:04:59.440
bar
0:04:56.880,0:05:01.600
all acting on the time independent
0:04:59.440,0:05:04.720
heisenberg state
0:05:01.600,0:05:07.280
okay so for operators we get this
0:05:04.720,0:05:07.280
relation
0:05:07.600,0:05:11.919
the time dependent operators in the
0:05:10.240,0:05:14.479
heisenberg picture
0:05:11.919,0:05:16.960
are just given by the time independent
0:05:14.479,0:05:19.840
operators in the schrodinger picture
0:05:16.960,0:05:22.479
pre and post multiplied by these unitary
0:05:19.840,0:05:22.479
operators
0:05:23.360,0:05:28.639
and this structure preserves
0:05:26.560,0:05:30.560
expectation values so let's take a
0:05:28.639,0:05:33.199
look at that
0:05:30.560,0:05:34.080
so the expectation value of an
0:05:33.199,0:05:37.280
operator
0:05:34.080,0:05:38.320
let's label it subscript S for now
0:05:37.280,0:05:39.199
to say that we're in the schrodinger
0:05:38.320,0:05:41.600
picture
0:05:39.199,0:05:42.479
sit's just
0:05:41.600,0:05:44.560
the operator
0:05:42.479,0:05:46.240
which is time independent sandwiched
0:05:44.560,0:05:49.280
between the states
0:05:46.240,0:05:52.639
psi which are a function of time but we
0:05:49.280,0:05:55.199
can write that as follows
0:05:52.639,0:05:56.560
where i've just taken the state psi in
0:05:55.199,0:05:57.039
the schrodinger picture as a function of
0:05:56.560,0:05:59.520
time
0:05:57.039,0:06:01.520
and written it as the unitary operator
0:05:59.520,0:06:04.400
acting on the time independent
0:06:01.520,0:06:05.600
heisenberg state and i've done the same
0:06:04.400,0:06:07.440
for the
0:06:05.600,0:06:09.680
hermitian conjugate over here and then
0:06:07.440,0:06:11.680
the Hermitian conjugate of this unitary
0:06:09.680,0:06:13.520
is this where the minus sign has
0:06:11.680,0:06:16.080
disappeared but then we just see
0:06:13.520,0:06:18.639
that this quantity here there's nothing
0:06:16.080,0:06:18.639
other than
0:06:18.720,0:06:26.560
the time dependent heisenberg
0:06:22.080,0:06:29.600
operator and so we can write this as
0:06:26.560,0:06:31.360
so it's just the expectation value
0:06:29.600,0:06:33.440
written in the heisenberg picture
0:06:31.360,0:06:34.560
and so this tells us is that whether we
0:06:33.440,0:06:35.600
work in the schrodinger picture or
0:06:34.560,0:06:37.680
heisenberg picture
0:06:35.600,0:06:38.720
expectation values are the same and of
0:06:37.680,0:06:39.120
course that must have been the case
0:06:38.720,0:06:41.039
because
0:06:39.120,0:06:42.720
expectation values are observable
0:06:41.039,0:06:45.840
quantities and they shouldn't depend on
0:06:42.720,0:06:45.840
our mathematical description
0:06:46.400,0:06:51.120
so let's return to our heisenberg
0:06:49.199,0:06:52.479
operator as a function of time
0:06:51.120,0:06:54.160
defined in terms of our time
0:06:52.479,0:06:56.319
independent Schroedinger operators
0:06:54.160,0:06:57.440
i should say i'm assuming here that
0:06:56.319,0:06:59.759
the schrodinger
0:06:57.440,0:07:01.199
operator has no time dependence we
0:06:59.759,0:07:02.479
assume that throughout the course if you
0:07:01.199,0:07:04.319
remember
0:07:02.479,0:07:05.840
one of the introductory videos i said
0:07:04.319,0:07:06.880
that our operators are always time
0:07:05.840,0:07:08.479
independent here
0:07:06.880,0:07:10.160
so our potential for example is always
0:07:08.479,0:07:12.160
time independent you can have an
0:07:10.160,0:07:13.680
explicit time dependence even in the
0:07:12.160,0:07:15.520
schrodinger picture
0:07:13.680,0:07:16.639
it doesn't make things too much more
0:07:15.520,0:07:17.680
complicated but we're not going to
0:07:16.639,0:07:20.319
consider that case
0:07:17.680,0:07:21.840
within this course so these operators
0:07:20.319,0:07:22.880
are always time independent here and the only
0:07:21.840,0:07:24.880
time dependence
0:07:22.880,0:07:26.160
in the heisenberg picture is coming in
0:07:24.880,0:07:31.840
in this form
0:07:26.160,0:07:31.840
and so we can evaluate the following
0:07:41.360,0:07:45.680
we can look at the derivative of this
0:07:43.919,0:07:47.360
operator with respect to time
0:07:45.680,0:07:49.199
and we know that the only time appearing
0:07:47.360,0:07:52.319
in this expression is here and here
0:07:49.199,0:07:52.319
and so we find the following
0:07:52.400,0:07:58.479
so we act first on
0:07:55.440,0:08:02.560
this part we bring down an i H over h
0:07:58.479,0:08:04.800
bar this commutes with this because
0:08:02.560,0:08:07.280
the exponential of this hamiltonian
0:08:04.800,0:08:09.599
operator with its pre-factors
0:08:07.280,0:08:10.879
is only a function of the hamiltonian
0:08:09.599,0:08:12.479
the hamiltonian always commutes with
0:08:10.879,0:08:13.199
itself and so it commutes with any power
0:08:12.479,0:08:15.120
of itself
0:08:13.199,0:08:16.479
and so it commutes with the exponential
0:08:15.120,0:08:17.759
of itself so
0:08:16.479,0:08:20.720
this could have been written here if we
0:08:20.720,0:08:24.240
then over here we've acted the time
0:08:22.720,0:08:26.160
derivative on this part
0:08:24.240,0:08:27.759
so we get the same thing this H can
0:08:26.160,0:08:30.960
equally well go over here
0:08:27.759,0:08:33.839
but it can't go over here because
0:08:30.960,0:08:35.519
H and the unknown operator
0:08:36.159,0:08:39.440
(we would know it it's just we haven't
0:08:37.680,0:08:40.880
said what it is) these two
0:08:39.440,0:08:44.640
probably won't commute and they may well
0:08:40.880,0:08:46.399
not so it must stay to the right of A_S
0:08:44.640,0:08:48.800
so if we look at what we have here this
0:08:46.399,0:08:52.320
thing here is just
0:08:48.800,0:08:54.320
A_H(t) by definition here we bring
0:08:52.320,0:08:57.120
the H over to this side and we get
0:08:54.320,0:08:57.120
A_H again
0:08:57.440,0:09:02.880
giving us this which we can rewrite
0:09:00.800,0:09:04.240
as follows where i've multiplied through
0:09:02.880,0:09:07.279
by
0:09:04.240,0:09:11.040
the h-bar and the i over here
0:09:07.279,0:09:13.760
so we've just got i h bar dA/dt
0:09:11.040,0:09:14.399
is equal to the commutator of A and H
0:09:13.760,0:09:16.160
and this
0:09:14.399,0:09:18.959
fulfills the role in the heisenberg
0:09:16.160,0:09:20.480
picture of the schrodinger equation
0:09:18.959,0:09:22.800
we call this the heisenberg equation of
0:09:20.480,0:09:30.240
motion so it's fulfilling the role of
0:09:22.800,0:09:30.240
the time-dependent schrodinger equation
V6.3 Ehrenfest's theorem
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continued from video V6.2) Ehrenfest's theorem, which specifies the time evolution of expectation values; obtaining classical equations of motion from the expectation values of quantum operators. Continued in video V6.4.
0:00:03.280,0:00:06.000
and we call this the heisenberg equation
0:00:04.720,0:00:07.359
of motion
0:00:06.000,0:00:09.519
so it's fulfilling the role of the
0:00:07.359,0:00:12.240
time-dependent schrodinger equation
0:00:09.519,0:00:13.599
if we take expectation values of
0:00:12.240,0:00:16.400
both sides
0:00:13.599,0:00:18.480
we can write this as follows where we've
0:00:16.400,0:00:21.840
sandwiched everything between
0:00:18.480,0:00:24.640
time independent heisenberg states
0:00:21.840,0:00:27.359
over here we of course just have the
0:00:24.640,0:00:28.960
expectation value
0:00:27.359,0:00:31.519
where again there's no need to write the
0:00:28.960,0:00:33.440
subscript H anywhere here because
0:00:31.519,0:00:34.800
expectation values are independent of
0:00:33.440,0:00:37.520
picture
0:00:34.800,0:00:39.120
and over here we can bring the states
0:00:37.520,0:00:42.239
within the time derivative because
0:00:39.120,0:00:42.239
they're time independent
0:00:42.640,0:00:46.399
but then we only have the expectation
0:00:44.160,0:00:48.800
value of A
0:00:46.399,0:00:50.480
sorry A operator it's written in
0:00:48.800,0:00:51.120
the Heisenberg picture but again expectation
0:00:50.480,0:00:52.879
values
0:00:51.120,0:00:55.520
are independent of picture and so we
0:00:52.879,0:00:58.559
have the result
0:00:55.520,0:01:01.280
i h bar d(expectation value of A)/dt
0:00:58.559,0:01:02.960
where A is an arbitrary operator
0:01:01.280,0:01:04.080
is equal to the expectation value of the
0:01:02.960,0:01:06.960
commutator of A
0:01:04.080,0:01:08.799
with the hamiltonian and this is a very
0:01:06.960,0:01:12.400
important result it's what's called
0:01:08.799,0:01:14.400
Ehrenfest's theorem
0:01:12.400,0:01:15.680
the reason it's so important is that it
0:01:14.400,0:01:17.119
gives us a connection between
0:01:15.680,0:01:19.920
quantum mechanics and classical
0:01:17.119,0:01:21.280
mechanics which from the very outset
0:01:19.920,0:01:22.080
it was known that there should be such a
0:01:21.280,0:01:24.640
connection
0:01:22.080,0:01:25.920
after all we were starting off with
0:01:24.640,0:01:27.200
classical systems and saying what
0:01:25.920,0:01:29.119
happens when we look at these on the
0:01:27.200,0:01:30.320
scale of individual particles
0:01:29.119,0:01:31.759
we'd like to think if we put enough
0:01:30.320,0:01:32.960
particles together we get the classical
0:01:31.759,0:01:36.130
result back again
0:01:32.960,0:01:37.600
and Ehrenfest's theorem tells us that
0:01:37.600,0:01:40.960
it's the expectation values of quantum
0:01:40.000,0:01:43.280
operators
0:01:40.960,0:01:44.320
which really behave like classical
0:01:43.280,0:01:46.399
objects
0:01:44.320,0:01:48.960
so in particular we can take a couple of
0:01:46.399,0:01:50.399
very important examples
0:01:48.960,0:01:52.840
so the first example let's take our
0:01:50.399,0:01:55.200
operator A is given by the position
0:01:52.840,0:01:56.560
operator
0:01:55.200,0:01:58.159
this is then the statement of
0:01:56.560,0:02:00.159
Ehrenfest's theorem we'll write the
0:01:58.159,0:02:02.640
hamiltonian
0:02:00.159,0:02:04.079
again as the sum of the kinetic term and
0:02:02.640,0:02:05.759
the potential term
0:02:04.079,0:02:08.399
but the potential term is always just a
0:02:05.759,0:02:10.399
function of the position operator
0:02:08.399,0:02:11.760
writing this all in terms of operators
0:02:10.399,0:02:13.599
and again any
0:02:11.760,0:02:16.319
function to be really recommend this as
0:02:13.599,0:02:18.160
some function of the position operator
0:02:16.319,0:02:19.680
where this is then defined by a taylor
0:02:18.160,0:02:22.879
series say of
0:02:19.680,0:02:23.520
this operator x. x will always commute
0:02:22.879,0:02:25.440
with any
0:02:23.520,0:02:28.879
function of x and so this is going to
0:02:25.440,0:02:31.920
disappear and we're just left with
0:02:28.879,0:02:36.720
the commutator of x with p squared
0:02:31.920,0:02:38.640
multiplied by 1 over 2m evaluate this we
0:02:36.720,0:02:41.840
have the commutator of x and p
0:02:38.640,0:02:45.200
and we use the result that
0:02:41.840,0:02:46.239
for matrices A and A,
0:02:45.200,0:02:48.640
[A,B^2]
0:02:46.239,0:02:50.000
is given by this which you can derive
0:02:48.640,0:02:52.080
fairly straightforwardly
0:02:50.000,0:02:53.840
and substituting it in and using our
0:02:52.080,0:02:57.680
canonical commutation relation
0:02:53.840,0:03:00.560
gives us this result i h bar
0:02:57.680,0:03:01.920
d by dt of the expectation value of the
0:03:00.560,0:03:05.360
position operator
0:03:01.920,0:03:05.760
is equal to i h bar over m multiplied
0:03:05.360,0:03:07.440
by
0:03:05.760,0:03:09.360
the expectation value of the position
0:03:07.440,0:03:12.720
operator momentum operator
0:03:09.360,0:03:14.879
we can cancel the ih bars to give the
0:03:12.720,0:03:18.640
result
0:03:14.879,0:03:19.920
dx by dt is p over m but of course this
0:03:18.640,0:03:21.280
is just the classical result this just
0:03:19.920,0:03:23.360
says that the velocity
0:03:21.280,0:03:24.400
is equal to the momentum divided by
0:03:23.360,0:03:26.400
the mass
0:03:24.400,0:03:28.400
so what Ehrenfest's theorem is showing
0:03:26.400,0:03:31.599
us is that
0:03:28.400,0:03:33.200
on average
0:03:31.599,0:03:34.560
where averages mean
0:03:33.200,0:03:35.360
the expectation value of the quantum
0:03:34.560,0:03:37.040
operators
0:03:35.360,0:03:38.959
on average we get back the classical
0:03:37.040,0:03:42.239
result let's take another look
0:03:38.959,0:03:43.120
at another example we take our operator
0:03:42.239,0:03:47.280
A is equal to p
0:03:43.120,0:03:49.519
so the equation says
0:03:47.280,0:03:50.400
whoops that seems to put me into a
0:03:49.519,0:03:52.799
different room
0:03:50.400,0:03:54.480
okay never mind so let's take the
0:03:52.799,0:03:56.799
example that the operator A
0:03:54.480,0:03:58.840
is equal to the momentum operator in
0:03:56.799,0:04:00.000
that case what we need to evaluate is
0:03:58.840,0:04:01.840
this
0:04:00.000,0:04:04.080
the commutator of the momentum with the
0:04:01.840,0:04:06.159
hamiltonian we can expand the hamiltonian
0:04:04.080,0:04:09.120
just as before
0:04:06.159,0:04:10.799
into the kinetic plus the potential part
0:04:09.120,0:04:13.840
the kinetic part will commute p
0:04:10.799,0:04:16.079
always commutes with p squared and we
0:04:13.840,0:04:19.199
need to evaluate the commutator of p
0:04:16.079,0:04:23.759
with the potential so the potential
0:04:19.199,0:04:26.240
is defined as a taylor series
0:04:23.759,0:04:28.160
that is it's just some function but it's
0:04:26.240,0:04:31.520
a function of the operator
0:04:28.160,0:04:33.840
the position operator x and
0:04:31.520,0:04:34.720
we can think of any such function of
0:04:33.840,0:04:36.320
an operator
0:04:34.720,0:04:39.120
as a taylor series in terms of that
0:04:36.320,0:04:41.360
operator so in particular we'd
0:04:39.120,0:04:43.440
like to work out what the commutator of
0:04:41.360,0:04:46.320
p with any different power
0:04:43.440,0:04:48.720
of the position operator x so let's work
0:04:46.320,0:04:50.960
with that separately
0:04:48.720,0:04:52.400
so we have that the commutator of our
0:04:50.960,0:04:53.280
momentum operator with our position
0:04:52.400,0:04:55.840
operator
0:04:53.280,0:04:58.320
is minus i h bar multiplied by the
0:04:55.840,0:05:00.960
identity operator
0:04:58.320,0:05:02.080
it's [p,x] so there's a minus sign
0:05:00.960,0:05:04.000
here
0:05:02.080,0:05:06.479
and if we write this out we get the
0:05:04.000,0:05:08.960
following expression
0:05:06.479,0:05:09.840
that is the p operator followed by the x
0:05:08.960,0:05:11.520
operator
0:05:09.840,0:05:14.320
is equal to the x operator followed by
0:05:11.520,0:05:16.400
the p operator we have to subtract
0:05:14.320,0:05:17.680
i h bar multiplied by the identity
0:05:16.400,0:05:20.240
operator from it
0:05:17.680,0:05:22.160
so this gives us a useful trick of pulling
0:05:20.240,0:05:24.400
one operator through another
0:05:22.160,0:05:26.000
so p and x don't commute so we can't say
0:05:24.400,0:05:28.080
that p x is x p
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we can pull the p through the x at the
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expense of adding in this extra term
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so for example we can look at the
0:05:31.680,0:05:35.520
commutator of p with x squared
0:05:35.680,0:05:40.600
which is p x^2 - x^2 p
0:05:38.479,0:05:41.680
and we can take a look at this term here
0:05:41.680,0:05:46.800
and realize that we have the following
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the p operator followed by the x
0:05:46.800,0:05:51.280
operator all followed by the x operator
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and we can use our expression from the
0:05:51.280,0:05:55.680
first equation
0:05:52.720,0:05:55.680
in parentheses here
0:05:55.759,0:06:02.800
to rewrite and expand this out
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to give this where we've used the fact
0:06:02.800,0:06:06.880
that the identity operator acting on
0:06:05.199,0:06:08.880
any operator just gives that operator
0:06:06.880,0:06:10.160
back so the identity acting on the
0:06:08.880,0:06:11.840
position operator gives the position
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operator
0:06:11.840,0:06:14.960
if we look at this expression we can do
0:06:13.440,0:06:16.880
the same trick again by noting that we
0:06:14.960,0:06:20.639
have a px here again
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we again expand it
0:06:20.639,0:06:24.639
and once again multiply out the
0:06:22.160,0:06:27.759
parentheses
0:06:24.639,0:06:28.720
so we have x squared p minus i h bar x
0:06:27.759,0:06:32.400
minus i h bar
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x these two combine into one
0:06:32.720,0:06:36.880
giving us x squared p minus two i h bar
0:06:35.360,0:06:40.160
x operator
0:06:36.880,0:06:42.240
and so overall we find the result
0:06:40.160,0:06:43.840
the commutator of p with x squared is
0:06:42.240,0:06:47.440
minus two i h bar
0:06:43.840,0:06:48.319
times x so we use this trick of
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pulling through
0:06:48.319,0:06:51.840
the momentum operator through the
0:06:49.599,0:06:52.720
position operator giving us an extra
0:06:51.840,0:06:54.560
term each time
0:06:52.720,0:06:56.800
because they don't commute now you can
0:06:54.560,0:06:57.440
do this repeatedly for higher powers
0:06:56.800,0:06:58.720
of x
0:06:57.440,0:07:00.800
and if you do it you find the following
0:06:58.720,0:07:03.840
result
0:07:00.800,0:07:07.440
commutator of p with x to the power n
0:07:03.840,0:07:08.400
is equal to minus n i h bar x to the n
0:07:07.440,0:07:10.240
minus 1.
0:07:08.400,0:07:11.520
so you see that what it is doing is very
0:07:10.240,0:07:14.800
much like a derivative
0:07:11.520,0:07:16.479
we have this commutator of p with x to
0:07:14.800,0:07:18.960
the n
0:07:16.479,0:07:20.080
lowering the power of x by one and
0:07:18.960,0:07:22.240
bringing the n out the front
0:07:20.080,0:07:25.599
it's a lot like a derivative but done
0:07:22.240,0:07:28.400
purely in terms of the operator algebra
0:07:25.599,0:07:28.880
next we need to write our potential
0:07:28.400,0:07:30.800
V(x)
0:07:28.880,0:07:33.199
out as a taylor series let's do that
0:07:30.800,0:07:33.199
over here
0:07:33.360,0:07:38.080
so we can write our potential V of the
0:07:36.560,0:07:40.240
operator x
0:07:38.080,0:07:41.520
as a taylor series we've got some
0:07:40.240,0:07:44.560
coefficients
0:07:41.520,0:07:46.319
a_n we'll separate out the n equals
0:07:44.560,0:07:47.759
zero term so we get a_0 times the
0:07:46.319,0:07:49.280
identity operator
0:07:47.759,0:07:51.680
and then sum from n equals one to
0:07:49.280,0:07:54.160
infinity of coefficient a_n
0:07:51.680,0:07:56.879
divided by n factorial multiplying x
0:07:54.160,0:07:58.080
operator to the power n
0:07:56.879,0:07:59.840
so it's a usual taylor series but
0:07:58.080,0:08:01.199
written in terms of operators and this
0:07:59.840,0:08:03.039
is what we mean when we say we have a
0:08:01.199,0:08:05.680
function of an operator
0:08:03.039,0:08:06.240
and now what we can do is we can take
0:08:05.680,0:08:08.639
the
0:08:06.240,0:08:12.080
commutator of this thing with momentum
0:08:08.639,0:08:14.000
operator p
0:08:12.080,0:08:15.759
is just to sum the taylor series so
0:08:14.000,0:08:17.440
we're just going to take the
0:08:15.759,0:08:19.360
commutator of p with each term
0:08:17.440,0:08:21.520
respectively the commutator
0:08:19.360,0:08:24.160
of any operator with the identity
0:08:21.520,0:08:26.800
operator is zero
0:08:24.160,0:08:28.160
and for each of the subsequent terms we
0:08:26.800,0:08:29.840
just use our previous relation then
0:08:28.160,0:08:33.599
we're going to drop a power of the
0:08:29.840,0:08:34.159
n from the power of x here
0:08:33.599,0:08:36.640
down
0:08:34.159,0:08:36.640
in front
0:08:37.599,0:08:43.839
pulling out the front the minus i h bar
0:08:40.000,0:08:43.839
and rewriting slightly we have
0:08:44.080,0:08:48.880
minus i h bar multiplying the sum from n
0:08:47.200,0:08:51.600
equals one to infinity a_n
0:08:48.880,0:08:52.000
over n minus one factorial multiplying x
0:08:51.600,0:08:54.560
to the n
0:08:52.000,0:08:56.080
minus one this is just some other
0:08:54.560,0:08:56.640
taylor series describing a different
0:08:56.080,0:08:58.560
function
0:08:56.640,0:09:00.000
but we know what function it is and you
0:08:58.560,0:09:01.600
can guess from the fact that
0:09:00.000,0:09:03.200
we have just dropped a power
0:09:01.600,0:09:05.360
here and the ns come out of the front
0:09:03.200,0:09:06.800
in fact this is now a good taylor series
0:09:05.360,0:09:07.680
to describe it the function we might
0:09:06.800,0:09:10.720
naturally call
0:09:07.680,0:09:13.920
V'(x)
0:09:10.720,0:09:15.200
that is there's a function V(x) and
0:09:13.920,0:09:15.920
we've taken the derivative of that
0:09:15.200,0:09:18.959
function
0:09:15.920,0:09:20.320
and then rather than just have it as
0:09:18.959,0:09:22.080
a function of the variable x we've had
0:09:20.320,0:09:24.160
as a function of operator x
0:09:22.080,0:09:26.720
and so this is the function V'
0:09:24.160,0:09:29.200
again just defined by its taylor series
0:09:26.720,0:09:30.880
and we're evaluating for the operator x
0:09:29.200,0:09:31.680
so the commutator with the momentum
0:09:30.880,0:09:34.640
operator
0:09:31.680,0:09:36.399
has brought out minus i h bar and
0:09:34.640,0:09:37.839
taken the derivative of the function
0:09:36.399,0:09:40.320
so that's what we need in our expression
0:09:37.839,0:09:40.320
over here
0:09:40.480,0:09:45.200
so we have that cancelling the ih bars
0:09:43.200,0:09:48.000
out
0:09:45.200,0:09:48.000
we get the result
0:09:48.399,0:09:52.399
the change in the expectation value of
0:09:50.800,0:09:53.120
the momentum operator with respect to
0:09:52.399,0:09:55.760
time
0:09:53.120,0:09:56.320
is equal to minus the expectation value
0:09:55.760,0:09:59.440
of
0:09:56.320,0:10:02.320
V' evaluated
0:09:59.440,0:10:04.320
for the position operator x where v
0:10:02.320,0:10:07.120
prime is the derivative of
0:10:04.320,0:10:09.519
the potential in three dimensions it
0:10:07.120,0:10:12.079
would take the following form
0:10:09.519,0:10:13.519
so in three dimensions it would be the
0:10:12.079,0:10:15.040
expectation value of the momentum
0:10:13.519,0:10:16.000
operator which is now a vector of
0:10:15.040,0:10:17.440
operators
0:10:16.000,0:10:19.200
the change in that with respect to
0:10:17.440,0:10:21.920
time is equal to minus
0:10:19.200,0:10:23.760
the expectation value of the gradient of
0:10:21.920,0:10:26.800
the potential
0:10:23.760,0:10:29.040
function evaluated for the operator x
0:10:26.800,0:10:30.640
so this then looks very much like
0:10:29.040,0:10:33.680
newton's second law
0:10:30.640,0:10:35.200
and it
0:10:33.680,0:10:35.760
would be very nice to say in an ideal
0:10:35.200,0:10:37.279
world
0:10:35.760,0:10:39.040
that while quantum mechanics and
0:10:37.279,0:10:41.760
classical mechanics are different
0:10:39.040,0:10:43.360
classical mechanics could be obeyed by
0:10:41.760,0:10:44.880
the expectation values of quantum
0:10:43.360,0:10:46.640
operators; that would be a nice
0:10:44.880,0:10:48.240
easy statement to make but unfortunately
0:10:46.640,0:10:51.040
it doesn't quite work like that
0:10:48.240,0:10:53.120
because for that to be true i'm going to
0:10:51.040,0:10:56.240
emphasize that this is not true
0:10:53.120,0:10:57.360
so this does not equal minus the
0:10:56.240,0:11:01.680
gradient
0:10:57.360,0:11:02.640
of v evaluated for the expectation value
0:11:01.680,0:11:04.399
of x
0:11:02.640,0:11:06.160
so this is what you need
0:11:04.399,0:11:07.920
to be true
0:11:06.160,0:11:09.760
for newton's second law to really be
0:11:07.920,0:11:11.279
returned
0:11:09.760,0:11:13.360
and and the statement to be the
0:11:11.279,0:11:15.600
expectation values of the
0:11:13.360,0:11:17.440
operators obey classical mechanics and
0:11:15.600,0:11:20.240
this is not true because in general
0:11:17.440,0:11:21.200
the expectation value of the function of
0:11:20.240,0:11:23.440
the operator
0:11:21.200,0:11:25.600
is not the same as the function of the
0:11:23.440,0:11:27.600
expectation value of the operator
0:11:25.600,0:11:29.360
the set of cases where that is true is
0:11:27.600,0:11:30.959
fairly small and we'll see some of those
0:11:29.360,0:11:32.560
later in the course
0:11:30.959,0:11:34.880
in particular when you have a quadratic
0:11:32.560,0:11:37.760
potential in position.
0:11:34.880,0:11:39.360
This is not the case and we can't
0:11:37.760,0:11:42.959
unfortunately make the statement
0:11:39.360,0:11:44.560
that the expectation values of
0:11:42.959,0:11:46.640
quantum operators of a classical
0:11:44.560,0:11:49.360
equation that's not quite true
0:11:46.640,0:11:51.680
instead Ehrenfest's theorem is often
0:11:49.360,0:11:53.360
cited as evidence in favor of the idea
0:11:51.680,0:11:55.040
of what's called the
0:11:53.360,0:11:56.399
correspondence principle
0:11:55.040,0:11:58.480
which says that classical mechanics
0:11:56.399,0:12:00.240
should be returned in the limit of large
0:11:58.480,0:12:02.880
quantum numbers so those are something
0:12:00.240,0:12:05.680
we'll see in a future video
0:12:02.880,0:12:06.720
but there's there's a way to look at
0:12:05.680,0:12:08.320
this mathematically
0:12:06.720,0:12:10.079
if you go into some detail you can
0:12:08.320,0:12:11.600
sort of get some evidence for an idea
0:12:10.079,0:12:13.279
as to how to get classical mechanics
0:12:11.600,0:12:14.399
back from quantum mechanics after all
0:12:13.279,0:12:17.279
you'd expect it to be
0:12:14.399,0:12:18.720
a kind of smooth limit but it's not
0:12:17.279,0:12:22.160
something we'll be taking a look at
0:12:18.720,0:12:24.160
just now in this course all right
0:12:22.160,0:12:27.360
so let's see if i can get myself back to
0:12:24.160,0:12:27.360
my usual room
0:12:27.440,0:12:30.880
so in the next video we're going to take
0:12:28.639,0:12:32.480
a look at applying the heisenberg
0:12:30.880,0:12:33.200
equation of motion to look at conserved
0:12:32.480,0:12:36.240
quantities
0:12:33.200,0:12:36.240
thank you for your time
V6.4 Conserved quantities
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continued from video V6.3) conserved quantities, whose expectation values are time independent; showing that this implies the corresponding operator commutes with the Hamiltonian; finding sets of operators which commute with the Hamiltonian, and assiging quantum numbers via their time-independent eigenvalues; the specific case of the quantum numbers of the electron in the Hydrogen atom.
0:00:00.480,0:00:04.319
hello in this video we're going to take
0:00:02.560,0:00:05.680
a look at conserved quantities
0:00:04.319,0:00:07.440
so we've seen in the previous couple of
0:00:05.680,0:00:09.840
videos that
0:00:07.440,0:00:12.880
Ehrenfest's theorem tells us that
0:00:09.840,0:00:16.080
expectation values of quantum operators
0:00:12.880,0:00:18.400
give us quantities associated with the
0:00:16.080,0:00:20.800
classical limits of quantum systems so
0:00:18.400,0:00:21.279
we get the classical equations of motion
0:00:20.800,0:00:24.080
back
0:00:21.279,0:00:25.279
on average where average means taking
0:00:24.080,0:00:28.320
the expectation value
0:00:25.279,0:00:30.800
of the corresponding operator so
0:00:28.320,0:00:32.559
if we'd like to look at conserved
0:00:30.800,0:00:33.760
quantities which is generally a sort of
0:00:32.559,0:00:34.800
thing we'd be interested in in physics
0:00:33.760,0:00:35.280
because these are things which don't
0:00:34.800,0:00:37.280
change
0:00:35.280,0:00:38.719
as a function of time it's reasonable to
0:00:37.280,0:00:41.520
assume that we might want them to obey
0:00:38.719,0:00:41.520
the following property
0:00:41.920,0:00:47.440
that is we have some quantum operator A
0:00:45.280,0:00:48.800
which will have a corresponding
0:00:47.440,0:00:50.640
observable quantity
0:00:48.800,0:00:52.320
given by its real eigenvalue because
0:00:50.640,0:00:53.920
this will be a Hermitian operator
0:00:52.320,0:00:55.600
we take the expectation value of that
0:00:53.920,0:00:57.280
for some state
0:00:55.600,0:00:59.039
and if the time derivative of that
0:00:57.280,0:01:00.800
quantity is equal to zero we call it a
0:00:59.039,0:01:03.600
conserved quantity and that should fit
0:01:00.800,0:01:05.119
our classical intuition
0:01:03.600,0:01:07.840
using Ehrenfest's theorem we can write
0:01:05.119,0:01:11.119
this as follows
0:01:07.840,0:01:13.840
so the commutator of the operator A
0:01:11.119,0:01:16.240
with the hamiltonian must be equal 0 for
0:01:13.840,0:01:19.439
this to be true
0:01:16.240,0:01:22.400
that is conserved quantities are
0:01:19.439,0:01:23.759
those physical observables whose quantum
0:01:22.400,0:01:26.320
Hermitian operators
0:01:23.759,0:01:26.960
commute with the hamiltonian and this is
0:01:26.320,0:01:30.000
assuming
0:01:26.960,0:01:32.079
as always in this course that the
0:01:30.000,0:01:33.520
operators including the hamiltonian do
0:01:32.079,0:01:35.520
not themselves have an explicit time
0:01:33.520,0:01:38.240
dependence
0:01:35.520,0:01:40.240
so if this is true just thinking of
0:01:38.240,0:01:41.600
these as matrices again which is true in
0:01:40.240,0:01:44.960
many cases otherwise
0:01:41.600,0:01:45.920
they're differential operators in either
0:01:44.960,0:01:48.000
case
0:01:45.920,0:01:49.360
if we have that the two things commute
0:01:48.000,0:01:49.920
that means that we should be able to
0:01:49.360,0:01:52.320
find
0:01:49.920,0:01:53.040
a set of simultaneous eigenvectors for
0:01:52.320,0:01:55.840
them
0:01:53.040,0:01:55.840
so we have the following
0:01:56.159,0:01:59.520
we know that this is the time
0:01:57.600,0:02:00.079
independent schrodinger equation so we
0:01:59.520,0:02:02.880
define
0:02:00.079,0:02:03.439
the eigen state of the hamiltonian to be
0:02:03.439,0:02:06.560
|n> which is labeled by this integer n
0:02:05.759,0:02:07.759
because then
0:02:06.560,0:02:10.239
we have the corresponding energy
0:02:07.759,0:02:12.319
eigenvalue E_n
0:02:10.239,0:02:14.080
and for our arbitrary operator A we
0:02:12.319,0:02:16.000
have
0:02:14.080,0:02:18.319
let's define this following equation so
0:02:16.000,0:02:19.120
the operator A has eigen states which
0:02:18.319,0:02:21.680
we'll label
0:02:19.120,0:02:22.319
|a> and we'll call their eigenvalues a as
0:02:21.680,0:02:24.160
well
0:02:22.319,0:02:25.599
as you've seen this is quite a common
0:02:24.160,0:02:28.640
notation that we tend to use
0:02:25.599,0:02:30.480
and so what we're saying is that if A
0:02:28.640,0:02:32.640
and H commute it's a general
0:02:30.480,0:02:33.519
theorem which you can prove in the
0:02:32.640,0:02:36.239
problem set
0:02:33.519,0:02:38.319
you expect that it's possible
0:02:36.239,0:02:40.480
to find a set of eigenvectors
0:02:38.319,0:02:42.000
which are simultaneously eigenvectors of
0:02:40.480,0:02:45.280
both of these operators
0:02:42.000,0:02:45.280
so we could have the following
0:02:45.599,0:02:48.959
where we've just written the state here
0:02:48.400,0:02:51.040
as
0:02:48.959,0:02:53.040
|n,a> so again this is just a
0:02:51.040,0:02:54.000
label we're just labeling our states in
0:02:53.040,0:02:56.000
our hilbert space
0:02:54.000,0:02:57.519
let's label it |n,a> but it's
0:02:56.000,0:02:59.680
suggestive because then
0:02:57.519,0:03:01.680
H acting on |n,a> returns the
0:02:59.680,0:03:04.959
eigenvalue E_n
0:03:01.680,0:03:07.200
multiplied by the state |n,a> and
0:03:04.959,0:03:08.159
the operator A acting on the same state
0:03:07.200,0:03:10.959
|n,a>
0:03:08.159,0:03:11.760
well that returns a|n,a>
0:03:11.760,0:03:17.040
okay so
0:03:15.040,0:03:18.800
it's a simultaneous eigenvector of both
0:03:17.040,0:03:20.959
of these operators
0:03:18.800,0:03:22.560
and then this quantity A and the
0:03:20.959,0:03:26.159
observable quantity
0:03:22.560,0:03:28.080
the real number a which its eigenvalue
0:03:26.159,0:03:31.040
will be conserved quantities they won't
0:03:28.080,0:03:31.040
be changing in time
0:03:31.120,0:03:35.040
so let's take a particular i look at a
0:03:33.599,0:03:37.760
particularly convenient example of this
0:03:35.040,0:03:39.280
a very important example
0:03:37.760,0:03:41.200
let's take the example that the operator
0:03:39.280,0:03:43.680
A is just the identity operator
0:03:41.200,0:03:45.680
which acts on any state and returns the
0:03:43.680,0:03:47.519
state itself
0:03:45.680,0:03:49.040
so clearly this commutes with the
0:03:47.519,0:03:52.000
hamiltonian because by definition it
0:03:49.040,0:03:52.000
commutes with everything
0:03:52.080,0:03:57.200
and so therefore the time derivative
0:03:55.200,0:04:00.000
of the expectation value of this for an
0:03:57.200,0:04:00.000
arbitrary state
0:04:00.319,0:04:04.640
zero but the expectation value of the
0:04:03.120,0:04:06.319
identity operator
0:04:04.640,0:04:10.480
is just the inner product of the state
0:04:06.319,0:04:10.480
with its Hermitian conjugate
0:04:10.720,0:04:14.879
and so we see that the inner product
0:04:13.680,0:04:17.440
of the state psi
0:04:14.879,0:04:19.199
with its Hermitian conjugate its time
0:04:17.440,0:04:20.799
derivative is zero
0:04:19.199,0:04:22.560
but this object here is nothing other
0:04:20.799,0:04:25.120
than our probability density
0:04:22.560,0:04:25.840
rewritten in terms of bras and
0:04:25.120,0:04:27.199
kets
0:04:25.840,0:04:29.360
so this is just the conservation of
0:04:27.199,0:04:31.040
probability
0:04:29.360,0:04:32.960
or in particular the global conservation
0:04:31.040,0:04:34.000
the total probability to
0:04:32.960,0:04:36.000
find the particle somewhere in the
0:04:34.000,0:04:38.400
universe is always equal to one
0:04:36.000,0:04:41.040
and so this follows let's take a look at
0:04:38.400,0:04:43.040
another example
0:04:41.040,0:04:45.120
so our Hamiltonian in general is written
0:04:43.040,0:04:47.919
as a sum of kinetic and potential
0:04:45.120,0:04:49.520
parts for the case of a free particle
0:04:47.919,0:04:50.639
by definition the potential part is
0:04:49.520,0:04:54.000
equal to zero
0:04:50.639,0:04:55.680
and in that case we have
0:04:54.000,0:04:58.160
H is the kinetic energy operator which
0:04:55.680,0:05:00.960
is just p squared over two m where p
0:04:58.160,0:05:01.680
is the momentum operator and so in this
0:05:00.960,0:05:04.479
case
0:05:01.680,0:05:05.759
we have that the momentum commutes
0:05:04.479,0:05:08.560
with the hamiltonian
0:05:05.759,0:05:11.840
because p always commutes with any
0:05:08.560,0:05:11.840
power of p
0:05:12.080,0:05:15.120
and so then in this case we can write
0:05:14.080,0:05:17.360
simultaneous
0:05:15.120,0:05:20.560
eigenvectors of p and H which we could
0:05:17.360,0:05:20.560
label as follows
0:05:20.960,0:05:24.160
where I've written the state with
0:05:22.639,0:05:25.600
the label |n,p>
0:05:24.160,0:05:28.400
because it's going to be an eigen state
0:05:25.600,0:05:28.720
of both the operator associated with
0:05:28.400,0:05:31.199
n
0:05:28.720,0:05:33.440
which is the hamiltonian H returning
0:05:31.199,0:05:36.639
eigenvalues E_n
0:05:33.440,0:05:39.680
and it's also an eigen state of p
0:05:36.639,0:05:41.680
so what this tells us is that we know
0:05:39.680,0:05:44.479
that whenever two operators commute
0:05:41.680,0:05:46.240
we can have simultaneous knowledge of
0:05:44.479,0:05:48.240
the corresponding observables
0:05:46.240,0:05:50.160
so in the case of a free particle we can
0:05:48.240,0:05:50.880
simultaneously know the energy of the
0:05:50.160,0:05:52.400
particle
0:05:50.880,0:05:53.919
and the momentum of the particle and
0:05:52.400,0:05:54.320
there's no contradiction there
0:05:54.320,0:05:57.919
the uncertainty relation doesn't hold
0:05:56.479,0:06:01.360
in that case because
0:05:57.919,0:06:02.880
or rather the
0:06:01.360,0:06:06.000
the probability of the uncertainties in
0:06:02.880,0:06:09.840
two states is zero
0:06:06.000,0:06:11.120
so in general it may not just be
0:06:09.840,0:06:12.880
one operator which commutes with the
0:06:11.120,0:06:13.919
hamiltonian we can have a larger set of
0:06:12.880,0:06:16.000
operators that can commute with the
0:06:13.919,0:06:18.880
hamiltonian
0:06:16.000,0:06:21.120
and it's important to try and work out
0:06:18.880,0:06:23.280
the maximal set of operators which
0:06:21.120,0:06:26.319
commute with a hamiltonian
0:06:23.280,0:06:28.400
because that set of operators we can
0:06:26.319,0:06:30.240
have simultaneous knowledge of
0:06:28.400,0:06:31.520
the corresponding physical observable
0:06:30.240,0:06:33.440
properties
0:06:31.520,0:06:34.720
and they'll give time-independent
0:06:33.440,0:06:37.440
expectation values so they'll be
0:06:34.720,0:06:39.840
conserved quantities
0:06:37.440,0:06:40.720
so the eigenvalues of operators which
0:06:39.840,0:06:43.759
commute with the
0:06:40.720,0:06:45.600
hamiltonian are called quantum numbers
0:06:43.759,0:06:47.520
and these are time independent
0:06:45.600,0:06:49.039
quantities that we can associate to the
0:06:47.520,0:06:53.520
hamiltonian
0:06:49.039,0:06:55.520
and they can all be known simultaneously
0:06:53.520,0:06:57.680
so let's look at a very important
0:06:55.520,0:07:00.000
example of this
0:06:57.680,0:07:01.280
so probably the most important example
0:07:00.000,0:07:03.680
we see in this course
0:07:01.280,0:07:05.360
are the states of the electron and the
0:07:03.680,0:07:07.919
hydrogen atom
0:07:05.360,0:07:10.080
so we can write these in ket notation as
0:07:07.919,0:07:13.280
follows
0:07:10.080,0:07:17.599
so we have as usual n the
0:07:13.280,0:07:19.840
quantum number associated with energy
0:07:17.599,0:07:22.160
in the hydrogen atom this is called
0:07:19.840,0:07:23.599
the principal quantum number
0:07:22.160,0:07:25.759
and it gives us the energy
0:07:23.599,0:07:27.520
eigenvalues as usual when acted on by
0:07:25.759,0:07:31.199
the hamiltonian
0:07:27.520,0:07:33.039
we can also extract l
0:07:31.199,0:07:34.560
so l is called the azimuthal quantum
0:07:33.039,0:07:36.880
number we'll see much more of it when we
0:07:34.560,0:07:40.560
come to study angular momentum
0:07:36.880,0:07:42.960
in lecture 10. and
0:07:40.560,0:07:45.440
it's associated with the operator which
0:07:42.960,0:07:48.800
is the square of the angular momentum
0:07:45.440,0:07:52.319
and it returns h bar squared l(l+1)
0:07:48.800,0:07:55.440
where l is an integer
0:07:52.319,0:07:59.039
greater than or equal to zero
0:07:55.440,0:08:02.240
we also have the quantum number
0:07:59.039,0:08:02.960
m which is called the magnetic quantum
0:08:02.240,0:08:06.639
number
0:08:02.960,0:08:09.599
and for the
0:08:06.639,0:08:10.479
z component of the angular
0:08:09.599,0:08:13.039
momentum
0:08:10.479,0:08:14.080
this operator returns the eigenvalue
0:08:13.039,0:08:17.280
h bar m
0:08:14.080,0:08:20.639
when acting on this state where m
0:08:17.280,0:08:22.639
is an integer which ranges from
0:08:20.639,0:08:23.919
-l to l
0:08:22.639,0:08:25.360
and again we'll see much more of this
0:08:23.919,0:08:26.400
when we look at angular momentum in more
0:08:25.360,0:08:28.080
detail
0:08:26.400,0:08:30.720
and finally we have s which we've seen
0:08:28.080,0:08:31.440
before which is just the spin quantum
0:08:30.720,0:08:32.719
number
0:08:31.440,0:08:34.479
where i've written it in slightly
0:08:32.719,0:08:36.399
strange notation here s is
0:08:34.479,0:08:37.839
plus or minus we could be referring to
0:08:36.399,0:08:40.959
say the z component
0:08:37.839,0:08:43.440
of this spin and it'll give us
0:08:40.959,0:08:44.880
eigenvalues plus or minus h bar over
0:08:43.440,0:08:48.240
2.
0:08:44.880,0:08:50.720
for spin up or spin down so
0:08:48.240,0:08:51.519
all of these different quantum numbers
0:08:50.720,0:08:53.680
can be
0:08:51.519,0:08:55.279
determined at the same time for the
0:08:53.680,0:08:58.959
electron and the hydrogen atom
0:08:55.279,0:09:01.360
so we have four quantum numbers
0:08:58.959,0:09:02.480
the
0:09:01.360,0:09:05.920
expectation values of
0:09:02.480,0:09:08.240
the corresponding operators will
0:09:05.920,0:09:10.240
be constant in time and so these quantum
0:09:08.240,0:09:12.720
numbers are well defined
0:09:10.240,0:09:15.440
for different times so they're sensible
0:09:12.720,0:09:19.200
to refer to as physical quantities
0:09:15.440,0:09:19.200
okay thank you for your time
V7.1 Infinite dimensional Hilbert spaces
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
unifying matrix mechanics and wave mechanics. Functions as infinite dimensional vectors; the position basis as a complete orthonormal basis; resolution of the identity into the position basis; inner products between complex functions; the wave function as the ket projected into the position basis; normalised states are represented by unit-length kets in Hilbert space.
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hello in this video we're going to take
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a look at
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infinite dimensional hilbert spaces or
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rather we're going to reassess what
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we've already done and see that we've
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actually already been working with
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infinite dimensional hilbert spaces
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when we've worked with functions so
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we've had two schemes that we've worked
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with the first is wave mechanics
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and second is matrix mechanics in the
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second we've had quantities like vectors
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and matrices
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and we're familiar with how to
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manipulate these things when
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the dimension of the space in which
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we're working has a finite number of
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dimensions
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in the case that we extend our space to
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have an infinite number of dimensions
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the objects just become the following:
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vectors become functions and matrices
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become differential operators
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and in general we can use the
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overarching terms
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'states', referring either to vectors in
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finite dimensional spaces or functions
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in infinite dimensional spaces, and
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'operators' referring to either matrices
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or differential operators so
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this might seem a little bit abstract
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thinking of a function say
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as an infinite dimensional vector but
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there's a couple of reasons that it's
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quite natural
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the first is if we think in terms of how
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a computer would display a function
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that is we would have to work with
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the discrete set of positions if the
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function is defined in position space
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which would be stored as a vector
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of different points and for each of
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those points we'd have a value of our
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function
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now if we want to approximate a
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smooth function
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taking a value in the in a real
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domain
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then we'd have to try and make the
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spacing of these points
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smaller and smaller something like this
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and as we take the limit of the number
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of points here going to infinity and the
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spacing between each one being
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infinitesimal
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we develop a smooth function
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but of course the computer only ever
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works with a discrete set of
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points along the real line it can't
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store
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an infinite set of numbers it only has a
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finite memory
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so in this sense you see that a function
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really is naturally an infinite
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dimensional vector
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because it has to take a value for each
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of an infinite number of
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real numbers of x another way to see
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in terms of matrices why this is is that
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if you think of
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we've been looking at eigenvalue
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equations from matrices
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and for an n by n matrix the
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eigenvalues are found by a
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characteristic polynomial
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and that characteristic polynomial
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has
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the form some coefficient times x
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plus another coefficient times x squared
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plus another coefficient times x cubed
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and so on up to x to the power of n
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so it's an nth order polynomial if we
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take the
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limit of the matrix becoming infinitely
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large the polynomial
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that describes the characteristic
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equation
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has to have an infinite number of terms
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an infinite number of different powers
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of x
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but if you write a polynomial which has
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an infinite number of powers of x in it
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like that what you've really written is
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a Taylor series which describes an
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arbitrary function
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and we can always expand a function of x
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as a taylor series like that or there's
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a
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large set of functions for which we can
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expand it as a Taylor series so
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your matrix then
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the characteristic polynomial stops
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being an nth order polynomial
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becomes an infinite order polynomial
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which is really just a function
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so it's quite natural to think of it in
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this way
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so in our finite dimensional vector
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spaces we can define
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orthonormal bases defined as follows
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that is sets of normalized vectors such
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that the inner product of the vector
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with itself will give one. This is the
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kronecker delta defined to be
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one when i equals j or zero when i
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doesn't equal j
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so if e_i and e_j are the same this is
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length one and if e_i and e_j
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are sorry i and j are different then
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it's equal to zero
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and this defines an orthonormal basis of
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vectors
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in the infinite dimensional space the
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equivalent to this is as follows
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we define an orthonormal basis
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of states x
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which are our position states and we say
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that
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if you have the inner product of x with
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y that's equal to
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the dirac delta function of x minus y
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this is in one dimension of space
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so remember the dirac delta function is
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really like a continuum limit of
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the kronecker delta
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it's equal to zero if x doesn't equal y
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and it equals infinity when x does equal
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y but in such a way that's an integral
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over it
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will give the value one so it's the
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natural generalization
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so whereas we have a finite number of
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vectors which span a finite dimensional
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vector space
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we must have an infinite number of these
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position vectors
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but again you can just think of this in
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terms of how a computer would store
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a set of real numbers it would really
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approximate them as a finite number of
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points
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so then it would be a finite dimensional
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vector space and these would be the
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points
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along that that space
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so another relation we have in the
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finite dimensional case is that the
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identity matrix
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which we can denote with this sort of
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blackboard bold
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one is equal to the sum from i equals
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one to n where this is the dimension of
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the space
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of the outer products of the
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normalized basis vectors in the
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infinite dimensional case the
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equivalent of this
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so the identity now which is really an
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identity operator
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is equal to the outer product of the
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position states integrated now
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over x from minus infinity to
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infinity
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so the integral is replacing the sum
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because we've taken the limit
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of the number of dimensions going to
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infinity
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in the finite dimensional case let's
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keep all the finite dimensional stuff to
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the side of the line
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we can express any vector in the space
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as a sum
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from y equals one to n then the size of
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the space the dimension of the space
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multiplying the basis vectors of the
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space
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by some coefficient of v_i and the
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coefficients v_i are given by the inner
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product
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of the basis vector with the vector v
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the equivalent to the infinite
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dimensional case is this
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so let's denote rather than choosing v
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for our finite dimensional vectors we'll
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choose
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f for our infinite dimensional ones
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because these are going to be functions
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and we can write them now as
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replace the sum with the integral
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so it's some function of x multiplying
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the basis states
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x and this function of x is defined to
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be
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the inner product of the x state
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so the basis vector with the function f
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so this takes a bit of getting used to
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but what we're saying here is that
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what we usually call
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our functions of x are really some more
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abstract
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concept they're an infinite dimensional
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vector projected into the x basis.
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So what other bases could they be
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projected into?
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Well actually they can be projected
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into other bases
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because we can write the function as a
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function of anything it doesn't have to
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be a function of position
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an example that we've actually seen
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already is that
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you can write your functions in momentum
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space and what you would do
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mathematically to write a function in
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position as a function of momentum is
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carry out a fourier transform
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and that's built in naturally into this
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structure so
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the function is actually a slightly more
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abstract entity than
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the thing that's a function of x the
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function could be a function of x or p
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or of anything so this ket notation
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indicates that the sort of true nature
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of the function
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before it's projected into a particular
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basis we'll see a concrete example of
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this in a second
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another key generalization we have
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is this
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so we have an inner product between our
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vectors
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and remember it's a complex vector space
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so
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writing the ket backwards like
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this -- the bra --
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is the hermitian conjugate
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and remember a Hilbert space which we're
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working with
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is defined to be a linear vector space
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but including an inner product
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and having all the vectors that we're
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referring to be normalizable
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so of finite length and this generalizes
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quite nicely in the basis of functions
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as follows
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so just as we can write the inner
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product of two vectors u and v
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like this we can write the inner product
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of two functions
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f and g in exactly the same way but to
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make it look
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more standard we can
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use the trick of inserting the
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identity okay so if we insert the
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identity between f and g
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we have the following we can always just
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put an identity
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between two vectors but then we use this
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identity up here
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to rewrite this as follows so an
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integral over dx
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and we've inserted the complete set of
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outer products here
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but then according to our definition up
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here
0:09:14.800,0:09:19.519
x inner products with g like this is
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just g of x
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and f inner product x well that has to
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be the hermitian conjugate sorry the
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complex conjugate
0:09:25.440,0:09:29.200
because this is just a complex number
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now because it's an inner product
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so this has to be the complex conjugate
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of x inner product f
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so it's as follows
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so the equivalent to the inner product
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in the vector space
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is the integral from minus infinity to
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infinity over x
0:09:44.160,0:09:49.040
of f*(x)g(x)
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okay so this is the inner product
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defined for functions in fact we've used
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that implicitly when we're working with
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wave mechanics earlier on in the course
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so the most important example of an
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infinite dimensional vector that we're
0:09:59.920,0:10:05.360
likely to encounter
0:10:01.360,0:10:06.000
is the following psi(x) the wave
0:10:05.360,0:10:07.040
function
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which we've been using in the wave
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mechanics part of the course
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in terms of matrix mechanics is just
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the ket psi projected into the x basis
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but we could equally well have projected
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it into the momentum basis
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in which case we would have psi of p
0:10:24.000,0:10:26.959
instead
0:10:24.880,0:10:28.720
so we sometimes write this as psi
0:10:26.959,0:10:29.040
twiddle and that's the fourier transform
0:10:28.720,0:10:32.399
of
0:10:29.040,0:10:34.480
psi of x okay so the ket
0:10:32.399,0:10:36.480
is really a slightly more general object
0:10:34.480,0:10:38.079
it's what lies at the heart of
0:10:36.480,0:10:40.160
the function without before you say it
0:10:38.079,0:10:41.200
lives in the x basis or the p basis or
0:10:40.160,0:10:42.240
whatever other way it says you want to
0:10:41.200,0:10:44.160
work
0:10:42.240,0:10:45.760
so a really important result we'll have
0:10:44.160,0:10:46.000
it's just re-expressing something we
0:10:45.760,0:10:49.519
already
0:10:46.000,0:10:52.640
know is as follows so we start from our
0:10:49.519,0:10:54.720
resolution of the identity
0:10:52.640,0:10:56.000
written in the position basis and
0:10:54.720,0:10:59.600
then we can sandwich this
0:10:56.000,0:11:01.839
with states side from the left and right
0:10:59.600,0:11:02.720
so I've just brought a bra psi from the
0:11:01.839,0:11:05.360
left, a
0:11:02.720,0:11:06.880
ket psi from the right and on this on
0:11:05.360,0:11:10.480
the right here i've done the same
0:11:06.880,0:11:12.880
the state psi is not itself a function
0:11:10.480,0:11:14.560
of x so it can go through the integral
0:11:12.880,0:11:16.000
it can be projected into the x basis
0:11:14.560,0:11:19.920
which is what's happening here
0:11:16.000,0:11:22.560
but the ket psi is not basis dependent
0:11:19.920,0:11:23.360
and so then this object is psi of x as
0:11:22.560,0:11:25.680
we've just said
0:11:23.360,0:11:28.640
and this object must be psi*(x)
0:11:25.680,0:11:29.839
the complex conjugate
0:11:28.640,0:11:32.399
so this is nothing other than the
0:11:29.839,0:11:36.000
modulus square of psi of x
0:11:32.399,0:11:37.680
and
0:11:36.000,0:11:40.480
the one can just disappear
0:11:37.680,0:11:42.320
because the identity operator acting on
0:11:40.480,0:11:44.240
a state is just a state so we have the
0:11:42.320,0:11:47.519
result
0:11:44.240,0:11:48.720
psi inner product psi is just equal to
0:11:47.519,0:11:51.120
the
0:11:48.720,0:11:53.360
integral over x of modulus psi squared
0:11:51.120,0:11:54.800
but we know that this thing equals one
0:11:53.360,0:11:57.040
this is just our normalization of the
0:11:54.800,0:11:58.560
wave function: the particle, while we
0:11:57.040,0:11:59.200
don't know where it exists, must exist
0:11:58.560,0:12:00.720
somewhere
0:11:59.200,0:12:02.320
and so the integral of the probability
0:12:00.720,0:12:03.040
density across all of space is equal to
0:12:02.320,0:12:06.560
one
0:12:03.040,0:12:08.160
and so we see that in terms of
0:12:06.560,0:12:10.800
vectors in terms of the infinite
0:12:08.160,0:12:11.760
dimensional Hilbert space expressed
0:12:10.800,0:12:15.600
as these states
0:12:11.760,0:12:17.680
we see that the state is of length 1.
0:12:15.600,0:12:19.920
so quantum states are complex vectors of
0:12:17.680,0:12:22.560
length 1 and vector here is used in the
0:12:19.920,0:12:23.519
more general sense where it could also
0:12:22.560,0:12:25.519
be
0:12:23.519,0:12:29.360
the ket form of a function before
0:12:25.519,0:12:29.360
projection into a particular basis
0:12:29.839,0:12:33.279
okay thank you for your time
V7.2 Fourier transforms
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
Fourier transforms written in terms of Dirac notation; Parseval's theorem; plane waves as eigenstates of the momentum operator.
0:00:00.399,0:00:05.040
hello in this video we're going to take
0:00:02.240,0:00:06.480
a look at fourier transforms
0:00:05.040,0:00:08.960
so we have the resolution of the
0:00:06.480,0:00:11.200
identity into the position basis
0:00:08.960,0:00:13.920
and we can also write it in the momentum
0:00:11.200,0:00:13.920
basis instead
0:00:14.480,0:00:20.480
so these states x and p
0:00:17.600,0:00:21.680
these are our complete
0:00:20.480,0:00:23.680
orthonormal bases
0:00:21.680,0:00:25.760
they're two different bases to describe
0:00:23.680,0:00:29.359
the same infinite dimensional
0:00:25.760,0:00:30.560
Hilbert space and we also have from
0:00:29.359,0:00:32.480
previous video
0:00:30.560,0:00:34.079
that they should be eigen states of the
0:00:32.480,0:00:34.800
position and momentum operators
0:00:34.079,0:00:38.719
respectively
0:00:34.800,0:00:40.480
that is we'd like the operators
0:00:38.719,0:00:43.600
x and p acting on their respective
0:00:40.480,0:00:45.600
eigenstates to return the eigenvalues
0:00:43.600,0:00:46.800
x and p so these things remember look a
0:00:45.600,0:00:50.239
little bit tautological
0:00:46.800,0:00:52.000
sorry i missed the corner of that ket
0:00:50.239,0:00:53.920
but they're different things operator
0:00:52.000,0:00:56.960
state eigenvalue
0:00:53.920,0:00:58.000
and the state again okay so we can use
0:00:56.960,0:01:00.160
this to show
0:00:58.000,0:01:02.000
the following result we have the
0:01:00.160,0:01:03.840
inner product of psi with itself must be
0:01:02.000,0:01:05.360
one
0:01:03.840,0:01:07.040
and finding this is our normalization
0:01:05.360,0:01:08.560
condition on the wave function
0:01:07.040,0:01:09.760
we can do exactly the same thing we can
0:01:08.560,0:01:10.720
start with this and we can insert
0:01:09.760,0:01:13.439
instead
0:01:10.720,0:01:15.360
a complete set of momentum states and we
0:01:13.439,0:01:16.880
find the following result
0:01:15.360,0:01:18.799
that one must also be equal to the
0:01:16.880,0:01:20.799
integral over all momenta
0:01:18.799,0:01:23.040
of the modulus square of the fourier
0:01:20.799,0:01:26.799
transform of the wave function
0:01:23.040,0:01:29.200
and so this relationship here that these
0:01:26.799,0:01:30.240
this quantity is unchanged whether
0:01:29.200,0:01:31.360
it's in x or p
0:01:30.240,0:01:34.400
there's an example of what's called
0:01:31.360,0:01:34.400
Parseval's theorem
0:01:34.640,0:01:39.280
so what we're saying is that x written
0:01:37.920,0:01:42.079
in the position basis
0:01:39.280,0:01:42.720
is related by fourier
0:01:42.079,0:01:46.320
transform
0:01:42.720,0:01:46.799
to psi written in the position
0:01:46.320,0:01:48.560
basis
0:01:46.799,0:01:50.880
is related to the fourier transform psi
0:01:48.560,0:01:52.000
written in the momentum basis
0:01:50.880,0:01:54.479
so let's look at that in a bit more
0:01:52.000,0:01:56.640
detail so we have that
0:01:54.479,0:01:58.320
the ket psi projected into the
0:01:56.640,0:02:01.119
position basis is the wave function
0:01:58.320,0:02:01.600
psi of x and then we can insert into
0:02:01.119,0:02:03.600
this
0:02:01.600,0:02:05.280
a complete set of momentum states as
0:02:03.600,0:02:07.200
follows
0:02:05.280,0:02:08.399
so i've just taken this object and i've
0:02:07.200,0:02:11.280
inserted
0:02:08.399,0:02:12.239
this integral p outer product p into the
0:02:11.280,0:02:14.800
middle between the
0:02:12.239,0:02:15.760
bra and the ket and then i've used the
0:02:14.800,0:02:18.560
fact that
0:02:15.760,0:02:18.879
x and in fact psi are not functions
0:02:18.560,0:02:20.640
of
0:02:18.879,0:02:22.319
momentum so we can pull it out in the
0:02:20.640,0:02:24.720
front here
0:02:22.319,0:02:25.440
and now we see that we have p inner
0:02:24.720,0:02:29.840
product psi
0:02:25.440,0:02:31.280
here which is psi twiddle of p
0:02:29.840,0:02:33.280
but now this is looking very much like a
0:02:31.280,0:02:35.040
fourier transform rather than inverse
0:02:33.280,0:02:36.800
fourier transform
0:02:35.040,0:02:41.040
but that would be true only if x inner
0:02:36.800,0:02:44.959
product p or equal to the following
0:02:41.040,0:02:47.680
e to the i p x over h bar where
0:02:44.959,0:02:49.360
because p x is not dimensionless it
0:02:47.680,0:02:51.040
has the units of h bar we should have a
0:02:49.360,0:02:52.400
dimensionless quantity up here so it's
0:02:51.040,0:02:55.200
natural to put e to the i p
0:02:52.400,0:02:56.319
x over h bar and then there's a
0:02:55.200,0:02:58.159
normalization on this
0:02:56.319,0:03:00.879
one over square root two pi h bar which
0:02:58.159,0:03:03.360
is convenient but there's this usual
0:03:00.879,0:03:04.879
ambiguous choice of normalization
0:03:03.360,0:03:06.239
when it comes to fourier transforms and
0:03:04.879,0:03:08.319
inverse fourier transforms
0:03:06.239,0:03:10.319
so let's use this definition so if we
0:03:08.319,0:03:12.000
say that x inner product p is this
0:03:10.319,0:03:14.560
then we have the usual fourier transform
0:03:12.000,0:03:14.560
written here
0:03:15.599,0:03:19.440
and the Dirac notation has naturally
0:03:17.040,0:03:23.440
encoded that fourier transform
0:03:19.440,0:03:26.080
in the following form x inner product p
0:03:23.440,0:03:27.760
is e to the i p x over h bar with a
0:03:26.080,0:03:30.239
normalization of one over
0:03:27.760,0:03:31.519
square root of two pi h bar and we
0:03:30.239,0:03:32.720
can check that the inv
0:03:31.519,0:03:34.159
sorry this is the inverse fourier
0:03:32.720,0:03:35.760
transform the fourier transform itself
0:03:34.159,0:03:38.319
works just as well
0:03:35.760,0:03:39.920
define p in a product psi is equal to
0:03:38.319,0:03:41.599
psi twiddle p
0:03:39.920,0:03:45.200
we can insert a complete set of
0:03:41.599,0:03:48.239
position states this time when we find
0:03:45.200,0:03:49.920
the following x inner product psi is psi of
0:03:48.239,0:03:51.840
x
0:03:49.920,0:03:54.239
and so this is a well-defined fourier
0:03:51.840,0:03:54.959
transform provided that this object p
0:03:54.239,0:03:58.080
inner product
0:03:54.959,0:03:58.080
x is given by
0:03:58.159,0:04:01.360
e to the minus i p x over h bar over
0:04:00.560,0:04:04.959
square root
0:04:01.360,0:04:07.200
2 pi h bar so this makes the usual
0:04:04.959,0:04:08.159
fourier transform / inverse fourier
0:04:07.200,0:04:10.159
transform pair
0:04:08.159,0:04:11.840
with the correct normalization and
0:04:10.159,0:04:15.280
you see that what we're saying here
0:04:11.840,0:04:16.959
is the following so we've had to use
0:04:15.280,0:04:18.160
the fact that p inner product x is the
0:04:16.959,0:04:21.919
complex conjugate
0:04:18.160,0:04:23.919
of x inner product p that is
0:04:21.919,0:04:25.280
but this is built into the direct
0:04:23.919,0:04:28.160
notation already that
0:04:25.280,0:04:29.520
for any vectors if we write x in the
0:04:28.160,0:04:31.440
product p
0:04:29.520,0:04:33.040
then p inner product x is the complex
0:04:31.440,0:04:34.080
conjugate of that that's the part of the
0:04:33.040,0:04:36.080
notation
0:04:34.080,0:04:37.360
and additionally we're saying that x
0:04:36.080,0:04:40.240
inner product p
0:04:37.360,0:04:41.520
is given by this form over here but we'd
0:04:40.240,0:04:43.280
also like to say that p
0:04:41.520,0:04:45.600
is an eigen state of the momentum
0:04:43.280,0:04:47.199
operator well what are eigen states of
0:04:45.600,0:04:49.840
the momentum operator
0:04:47.199,0:04:50.479
they're nothing other than plane waves
0:04:49.840,0:04:51.840
so
0:04:50.479,0:04:54.639
what we're saying we'd like to say is
0:04:51.840,0:04:55.280
that x inner product p is a projection
0:04:54.639,0:04:58.720
of
0:04:55.280,0:05:00.960
the plane wave into the position basis
0:04:58.720,0:05:03.039
and that's exactly what this is right
0:05:00.960,0:05:05.919
this is the form of a plane wave
0:05:03.039,0:05:06.560
in this case a right-going plane wave
0:05:05.919,0:05:08.880
in this case
0:05:06.560,0:05:10.880
a left-going plane wave the complex
0:05:08.880,0:05:12.479
conjugate
0:05:10.880,0:05:14.000
so all of these structures worked out
0:05:12.479,0:05:17.280
quite nicely and naturally
0:05:14.000,0:05:19.039
in the in the Dirac notation so
0:05:17.280,0:05:20.160
fourier transforms all work very well
0:05:19.039,0:05:22.160
and there's questions about this in the
0:05:20.160,0:05:23.840
problem set
0:05:22.160,0:05:25.680
so just to finish up we'll look at that
0:05:23.840,0:05:27.039
in slightly more detail so writing in
0:05:25.680,0:05:29.520
the position basis
0:05:27.039,0:05:30.320
we've seen already that the momentum
0:05:29.520,0:05:33.440
operator
0:05:30.320,0:05:33.440
is expressed as follows
0:05:33.600,0:05:39.680
and so the claim is that the
0:05:37.120,0:05:40.720
plane waves are the eigen states of this
0:05:39.680,0:05:42.160
operator
0:05:40.720,0:05:44.720
and we can check that straightforwardly
0:05:42.160,0:05:46.800
by substituting
0:05:44.720,0:05:48.080
so insert the plane wave from the
0:05:46.800,0:05:49.600
previous board
0:05:48.080,0:05:52.960
insert the form of the operator it
0:05:49.600,0:05:54.720
brings down an ip over h bar
0:05:52.960,0:05:56.720
and so we see that we have the following
0:05:54.720,0:05:58.960
result
0:05:56.720,0:05:59.919
p acting on the plane wave state is
0:05:58.960,0:06:03.120
equal to
0:05:59.919,0:06:04.960
the eigenvalue a real number acting
0:06:03.120,0:06:06.240
again on the plane wave state and so
0:06:04.960,0:06:07.600
then it's natural to interpret this
0:06:06.240,0:06:11.360
object
0:06:07.600,0:06:12.160
as the eigen state of the momentum
0:06:11.360,0:06:13.520
operator
0:06:12.160,0:06:16.800
but clearly it's written in the position
0:06:13.520,0:06:18.400
basis and so it's really this
0:06:16.800,0:06:20.560
and so that's what we saw in the
0:06:18.400,0:06:21.440
previous board okay so in the next video
0:06:20.560,0:06:24.479
we'll take a look
0:06:21.440,0:06:28.720
in a bit more detail at operators
0:06:24.479,0:06:28.720
in this formalism thanks for your time
V7.3 Differential operators
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
differential operators as the generalisation of matrices to infinite dimensional Hilbert spaces. Representations of common operators in the position and momentum bases; expectation values of powers of the position and momentum operators; the Hermiticity of differential operators.
0:00:00.160,0:00:05.279
hello in this video we're going to take
0:00:02.240,0:00:09.840
a look at differential operators
0:00:05.279,0:00:12.320
that is the equivalent to matrices
0:00:09.840,0:00:14.799
in a finite dimensional vector space but
0:00:12.320,0:00:17.359
in an infinite dimensional vector space
0:00:14.799,0:00:19.840
so we can construct the following table
0:00:17.359,0:00:22.640
of relations
0:00:19.840,0:00:23.039
we have our general operator something
0:00:23.039,0:00:26.560
like the following we can have the
0:00:24.800,0:00:27.680
position operator which we've already
0:00:26.560,0:00:31.199
seen
0:00:27.680,0:00:31.840
or the momentum operator or the energy
0:00:31.199,0:00:36.559
operator
0:00:31.840,0:00:39.840
H the hamiltonian which is equal to
0:00:36.559,0:00:42.840
the momentum operator squared
0:00:39.840,0:00:44.960
over 2m plus the potential energy
0:00:42.840,0:00:48.160
operator and we sometimes
0:00:44.960,0:00:51.199
have been calling this term T the
0:00:48.160,0:00:54.399
kinetic energy operator
0:00:51.199,0:00:56.559
and written out in terms of when we're
0:00:54.399,0:00:57.680
doing Schrodinger wave mechanics
0:00:56.559,0:00:59.600
we're writing things in terms of
0:00:57.680,0:01:01.760
functions which are then
0:00:59.600,0:01:03.520
infinite dimensional vectors in our
0:01:01.760,0:01:06.560
complex Hilbert spaces
0:01:03.520,0:01:09.600
and so we have the following forms
0:01:06.560,0:01:12.720
so writing these out as
0:01:09.600,0:01:16.080
represented in the position basis the
0:01:12.720,0:01:19.200
x operator is just simply
0:01:16.080,0:01:20.720
the position x the real number
0:01:19.200,0:01:22.640
the momentum operator on the other hand
0:01:20.720,0:01:23.119
is a bit more interesting it takes the
0:01:22.640,0:01:26.720
form
0:01:23.119,0:01:28.640
minus i h bar d by dx
0:01:26.720,0:01:30.560
as we've seen a couple of times now and
0:01:28.640,0:01:32.240
this is in one dimension
0:01:30.560,0:01:34.880
and then the hamiltonian takes the
0:01:32.240,0:01:37.759
following form
0:01:34.880,0:01:39.520
it's p squared over 2m p is defined here
0:01:37.759,0:01:40.400
and so it must be minus hbar squared
0:01:39.520,0:01:44.159
over two m
0:01:40.400,0:01:46.720
d by dx squared and the
0:01:44.159,0:01:48.000
potential operator in the position basis
0:01:46.720,0:01:50.240
is just the potential
0:01:48.000,0:01:51.759
as a function of x out of interest
0:01:50.240,0:01:52.320
although it won't be a key focus in this
0:01:51.759,0:01:53.520
course
0:01:52.320,0:01:56.960
we could also write things in the
0:01:53.520,0:01:58.880
position momentum basis
0:01:56.960,0:02:01.680
so first the momentum operator in this
0:01:58.880,0:02:04.000
case takes the trivial form
0:02:01.680,0:02:05.200
p just the real number much as x did in
0:02:04.000,0:02:09.039
the x
0:02:05.200,0:02:14.000
basis the position operator now
0:02:09.039,0:02:16.319
takes the form minus i h bar d by d p
0:02:14.000,0:02:18.080
and the hamiltonian now must take the
0:02:16.319,0:02:19.920
form
0:02:18.080,0:02:21.599
p squared over two m where p is just the
0:02:19.920,0:02:25.120
real number p
0:02:21.599,0:02:27.680
and V now has to be a function of
0:02:25.120,0:02:27.680
excuse me
0:02:28.480,0:02:32.160
V the potential is now a function of
0:02:30.879,0:02:34.560
minus i h bar d
0:02:32.160,0:02:35.440
by d p where this is now a differential
0:02:34.560,0:02:38.000
operator
0:02:35.440,0:02:40.959
acting on functions of p and is defined
0:02:38.000,0:02:42.560
by its taylor series
0:02:40.959,0:02:44.959
so we've seen a little bit already of
0:02:42.560,0:02:46.720
expectation values of operators
0:02:44.959,0:02:49.120
in particular we know that they take the
0:02:46.720,0:02:51.519
following form
0:02:49.120,0:02:53.040
so the expectation value of the operator
0:02:51.519,0:02:56.080
A according to
0:02:53.040,0:02:58.720
in the state psi is just a
0:02:56.080,0:03:00.000
sandwich between the bracket of psi with
0:02:58.720,0:03:03.360
itself
0:03:00.000,0:03:04.080
so a particular set of operators
0:03:03.360,0:03:06.319
we're very interested
0:03:04.080,0:03:07.760
in are powers of the position of
0:03:06.319,0:03:09.120
momentum operators in fact that's
0:03:07.760,0:03:10.239
basically everything that we ever really
0:03:09.120,0:03:12.480
want to look at
0:03:10.239,0:03:13.920
in this course at least i'm not
0:03:12.480,0:03:15.040
really familiar with any situations in
0:03:13.920,0:03:18.959
quantum mechanics where you want to look
0:03:15.040,0:03:21.360
at expectation values i'm just talking
0:03:18.959,0:03:22.959
and then say we'd like to find the
0:03:21.360,0:03:26.959
expectation value of
0:03:22.959,0:03:30.159
the position operator to the power n
0:03:26.959,0:03:34.000
well we can act that in from the left
0:03:30.159,0:03:35.599
as follows so the position operator
0:03:34.000,0:03:36.239
to the power n acting on the identity
0:03:35.599,0:03:37.599
operator
0:03:36.239,0:03:39.280
is just the position operator to the
0:03:37.599,0:03:40.959
power n the density operator is like the
0:03:39.280,0:03:42.720
'one' of operators
0:03:40.959,0:03:44.480
and over here we've brought it
0:03:42.720,0:03:46.000
through the integral which might look a
0:03:44.480,0:03:48.080
bit dodgy because this is integrating
0:03:46.000,0:03:49.040
over x but this is remember this is the
0:03:48.080,0:03:52.159
operator
0:03:49.040,0:03:52.879
x to the n rather than the
0:03:52.159,0:03:55.280
eigenvalue
0:03:52.879,0:03:56.640
x which this is integrating over so it's
0:03:55.280,0:03:57.920
actually fine to bring that through the
0:03:56.640,0:04:02.159
integral here
0:03:57.920,0:04:04.400
but then x operator acting on state x
0:04:02.159,0:04:07.519
this is by definition the eigenstate
0:04:04.400,0:04:10.879
of the position operator and so
0:04:07.519,0:04:14.080
we can act this n times and we just
0:04:10.879,0:04:17.040
bring down n powers of the
0:04:14.080,0:04:18.160
the eigenvalue associated with that
0:04:17.040,0:04:20.079
so it's x to the n
0:04:18.160,0:04:22.160
and now it's trapped inside the integral
0:04:20.079,0:04:22.639
here and can't be taken out because this
0:04:22.160,0:04:23.840
is now
0:04:22.639,0:04:26.479
in fact the thing that's being
0:04:23.840,0:04:29.040
integrated so
0:04:26.479,0:04:30.000
we'd like to evaluate the expectation
0:04:29.040,0:04:33.759
value of this
0:04:30.000,0:04:34.560
operator and so what we need to do is
0:04:33.759,0:04:36.800
sandwich it
0:04:34.560,0:04:38.800
between states psi which again can fit
0:04:36.800,0:04:42.479
in through the integral here
0:04:38.800,0:04:44.960
and we get the following result
0:04:42.479,0:04:46.080
so we have x inner product psi which is
0:04:44.960,0:04:49.120
psi of x
0:04:46.080,0:04:50.400
the x to the n can just pull out
0:04:49.120,0:04:51.600
through here it doesn't do anything to
0:04:50.400,0:04:54.720
these states
0:04:51.600,0:04:56.560
and so we get the final result the
0:04:54.720,0:04:58.400
expectation value of the x position
0:04:56.560,0:05:00.639
operators the raised to the power n
0:04:58.400,0:05:02.800
is equal to the integral over all of x
0:05:00.639,0:05:04.720
of the modulus of psi squared
0:05:02.800,0:05:06.720
times x to the n now this makes sense in
0:05:04.720,0:05:10.160
terms of statistics
0:05:06.720,0:05:11.759
because modulus psi x squared is the
0:05:10.160,0:05:13.440
probability density
0:05:11.759,0:05:15.360
so if we just integrate that by itself
0:05:13.440,0:05:17.199
we get one
0:05:15.360,0:05:19.120
but if we wanted to evaluate
0:05:17.199,0:05:22.560
expectation values of different
0:05:19.120,0:05:24.880
quantities in statistics we would
0:05:22.560,0:05:26.400
average those things weighted by the
0:05:24.880,0:05:27.520
probability distribution
0:05:26.400,0:05:29.840
and that's exactly what we're finding
0:05:27.520,0:05:32.240
here in quantum mechanics as well
0:05:29.840,0:05:34.880
to find the expectation value of
0:05:32.240,0:05:37.199
momentum operators raised to the power n
0:05:34.880,0:05:38.880
we can do exactly the same thing using
0:05:37.199,0:05:41.759
the resolution of the identity into
0:05:38.880,0:05:41.759
momentum states
0:05:42.080,0:05:45.360
and all the working works as before the
0:05:44.160,0:05:46.639
momentum operator actually in the
0:05:45.360,0:05:48.560
momentum eigenstate
0:05:46.639,0:05:49.759
gives the momentum eigenvalue we
0:05:48.560,0:05:52.320
sandwich it between states
0:05:49.759,0:05:52.960
psi and so we arrive at the same
0:05:52.320,0:05:54.080
expression
0:05:52.960,0:05:55.759
but everything written in terms of
0:05:54.080,0:05:56.560
momentum rather than position again
0:05:55.759,0:05:58.560
there's this
0:05:56.560,0:06:00.240
equivalent between writing things in
0:05:58.560,0:06:03.600
terms of position and momentum
0:06:00.240,0:06:05.600
in quantum mechanics okay so
0:06:03.600,0:06:07.120
let's look at the hermeticity of
0:06:05.600,0:06:07.600
these operators because when we looked
0:06:07.120,0:06:10.400
at
0:06:07.600,0:06:11.280
finite dimensional Hilbert
0:06:10.400,0:06:14.800
spaces
0:06:11.280,0:06:16.400
we said that our matrices that we were
0:06:14.800,0:06:17.759
using had to be Hermitian
0:06:16.400,0:06:21.840
so there should be an equivalent of that
0:06:17.759,0:06:21.840
statement for differential operators
0:06:22.080,0:06:26.240
so i'm just going to state the
0:06:24.560,0:06:29.360
general expression for finding if an
0:06:26.240,0:06:31.919
operator is hermitian or not
0:06:29.360,0:06:32.800
so the operator A is Hermitian A equals
0:06:31.919,0:06:36.319
A^dagger
0:06:32.800,0:06:38.400
if and only if this statement here holds
0:06:36.319,0:06:39.919
the integral of the operator acting on
0:06:38.400,0:06:42.080
phi of x
0:06:39.919,0:06:43.520
complex conjugate right because this
0:06:42.080,0:06:46.479
is a differential operator
0:06:43.520,0:06:48.160
this is a complex function so then this
0:06:46.479,0:06:48.639
thing must be a complex function so it's
0:06:48.160,0:06:50.800
the
0:06:48.639,0:06:51.919
complex conjugate not the Hermitian
0:06:50.800,0:06:54.639
conjugate
0:06:51.919,0:06:55.759
multiplying psi of x into greatest
0:06:54.639,0:06:58.240
overall of x
0:06:55.759,0:06:59.520
is equal to the complex conjugate of phi
0:06:58.240,0:07:01.759
of x
0:06:59.520,0:07:03.039
multiplying the operator acting on psi
0:07:01.759,0:07:05.599
of x integrated
0:07:03.039,0:07:06.319
over x and that has to hold true for all
0:07:05.599,0:07:09.520
arbitrary
0:07:06.319,0:07:11.840
phi of x and psi of x complex functions
0:07:09.520,0:07:13.440
okay so it's best to just take it as a
0:07:11.840,0:07:15.120
definition you can derive it and it's
0:07:13.440,0:07:16.960
not too complicated
0:07:15.120,0:07:18.160
but it's beyond the scope of this
0:07:16.960,0:07:20.400
course
0:07:18.160,0:07:21.599
so let's take a look at a couple of
0:07:20.400,0:07:24.160
important cases
0:07:21.599,0:07:25.680
so the first hermitian operator
0:07:26.319,0:07:32.720
is the position operator A is equal to x
0:07:30.560,0:07:34.160
so we just need to substitute it into
0:07:32.720,0:07:35.120
this form and check we can get it into
0:07:34.160,0:07:37.199
that form
0:07:35.120,0:07:38.479
and that's quite trivial in this case
0:07:37.199,0:07:41.440
so the left hand side
0:07:38.479,0:07:42.800
gives this but the position operator
0:07:41.440,0:07:46.000
in the position basis
0:07:42.800,0:07:47.440
is just the position this is written by
0:07:46.000,0:07:48.800
the way in the position basis of course
0:07:47.440,0:07:50.560
we could have written it in the momentum
0:07:48.800,0:07:54.400
basis if we wanted to
0:07:50.560,0:07:55.599
so but the position is a real
0:07:54.400,0:07:58.560
number we know
0:07:55.599,0:07:59.680
and so we can bring down the complex
0:07:58.560,0:08:02.879
conjugates here
0:07:59.680,0:08:02.879
it doesn't affect x
0:08:03.199,0:08:07.360
and we can happily bring the x over to
0:08:05.120,0:08:07.360
here
0:08:09.120,0:08:15.199
and finally the eigenvalue x
0:08:12.240,0:08:16.639
multiplying the function psi of x
0:08:15.199,0:08:20.000
could equally well have been written
0:08:16.639,0:08:22.080
operator x because the
0:08:20.000,0:08:23.599
x operator acting on the function of x
0:08:22.080,0:08:25.440
will just turn into x
0:08:23.599,0:08:27.280
acting on the function of x and so we've
0:08:25.440,0:08:29.759
proven the right hand side
0:08:27.280,0:08:31.120
and so the operator x is hermitian
0:08:29.759,0:08:33.039
and that's good news because
0:08:31.120,0:08:34.479
that means that the eigenstates of
0:08:33.039,0:08:35.839
this operator are real
0:08:34.479,0:08:37.839
and those are of course our positions
0:08:35.839,0:08:40.240
we'd like our positions to be
0:08:37.839,0:08:41.519
measurable and we'd like to be real so
0:08:40.240,0:08:44.959
slightly more tricky case
0:08:41.519,0:08:46.480
is the momentum operator
0:08:44.959,0:08:48.959
and we get this expression where i've
0:08:46.480,0:08:50.560
used the form minus i h bar d by dx
0:08:48.959,0:08:52.160
for the momentum operator in the
0:08:50.560,0:08:54.959
position basis
0:08:52.160,0:08:56.720
the complex conjugate of this thing
0:08:54.959,0:08:59.839
it changes the sign of this part
0:08:56.720,0:09:02.080
and a complex conjugates the function
0:08:59.839,0:09:02.080
phi
0:09:04.000,0:09:07.120
so how do we get the operator onto
0:09:06.399,0:09:10.640
this part
0:09:07.120,0:09:12.480
where we can use integration by parts
0:09:10.640,0:09:14.480
so we know this expression equals the
0:09:12.480,0:09:18.959
following
0:09:14.480,0:09:22.080
so it equals the integral
0:09:18.959,0:09:23.839
evaluates at the limits minus sticking
0:09:22.080,0:09:26.959
the derivative on the other part
0:09:23.839,0:09:30.560
so we have so
0:09:26.959,0:09:33.040
we've got a minus by h bar d by dx here
0:09:30.560,0:09:36.800
so this part is just p acting on psi
0:09:33.040,0:09:39.839
which is what we'd like let's write that
0:09:36.800,0:09:40.720
so the left hand side equals the right
0:09:39.839,0:09:43.519
hand side
0:09:40.720,0:09:45.760
provided this term equals zero but this
0:09:43.519,0:09:49.200
must equal zero because we're evaluating
0:09:45.760,0:09:50.640
phi star of x and psi of x
0:09:49.200,0:09:52.720
but these are functions which live in
0:09:50.640,0:09:53.279
our infinite dimensional complex hilbert
0:09:52.720,0:09:54.640
space
0:09:53.279,0:09:56.800
and remember it's a definition of the
0:09:54.640,0:09:59.600
hilbert space that the states be
0:09:56.800,0:10:01.279
square normalizable so if we take the
0:09:59.600,0:10:03.040
modulus square of any state
0:10:01.279,0:10:05.920
and evaluate it at plus or minus
0:10:03.040,0:10:08.959
infinity it must go to zero
0:10:05.920,0:10:10.079
because otherwise we wouldn't be
0:10:08.959,0:10:11.519
able to carry out a normalization
0:10:10.079,0:10:14.959
condition that the integral of
0:10:11.519,0:10:16.640
modulus psi squared over minus infinity
0:10:14.959,0:10:19.120
to infinity is equal to one
0:10:16.640,0:10:20.399
so actually these must be zero for our
0:10:19.120,0:10:22.160
states to be normalizable
0:10:20.399,0:10:23.600
and so we've proven the hermiticity of
0:10:22.160,0:10:26.480
the momentum operator
0:10:23.600,0:10:28.480
is written as a differential operator so
0:10:26.480,0:10:31.839
again that's good news
0:10:28.480,0:10:31.839
okay thank you for your time
V7.4 The postulates of quantum mechanics
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
the postulates of quantum mechanics.
0:00:00.000,0:00:04.160
hello in this video we're going to take
0:00:02.000,0:00:04.880
a look at the postulates of quantum
0:00:04.160,0:00:05.920
mechanics
0:00:04.880,0:00:07.520
and they're all going to fit on this
0:00:05.920,0:00:08.480
board and they really define the whole
0:00:07.520,0:00:11.599
subject
0:00:08.480,0:00:12.639
so it's taken us this many videos to get
0:00:11.599,0:00:14.080
to this point because
0:00:12.639,0:00:15.679
you needed a lot of background to be
0:00:14.080,0:00:17.039
able to state them but now we can state
0:00:15.679,0:00:20.720
them you'll you should know all of them
0:00:17.039,0:00:22.640
already but let's take a look at them
0:00:20.720,0:00:24.080
the states of the system are represented
0:00:22.640,0:00:26.000
by ket psi
0:00:24.080,0:00:30.320
in a complex hilbert space which we'll
0:00:26.000,0:00:32.719
call curly h
0:00:30.320,0:00:33.520
quantities are represented by hermitian
0:00:32.719,0:00:35.280
operators
0:00:33.520,0:00:36.960
and we'll call them capital A with a hat
0:00:35.280,0:00:38.879
on them which are
0:00:36.960,0:00:40.320
living in the hilbert space so if the
0:00:38.879,0:00:41.440
Hilbert space is finite dimensional
0:00:40.320,0:00:43.040
these would be matrices
0:00:41.440,0:00:45.840
if it's infinite dimensional they'll be
0:00:43.040,0:00:47.760
differential operators
0:00:45.840,0:00:50.239
all such operators are assumed to
0:00:47.760,0:00:53.680
possess a complete set of eigen states
0:00:50.239,0:00:55.199
that is the operator A acting on
0:00:53.680,0:00:58.640
eigenstate |a_n>
0:00:55.199,0:01:00.719
returns eigenvalue a_n
0:00:58.640,0:01:02.800
multiplying the eigenstate the important
0:01:00.719,0:01:04.559
bit here that needs to be postulated
0:01:02.800,0:01:06.080
is the fact that it forms a complete set
0:01:04.559,0:01:09.280
of eigenstates so this
0:01:06.080,0:01:12.000
may not be true in the general case
0:01:09.280,0:01:15.119
but it's
0:01:12.000,0:01:17.360
always true in quantum mechanics
0:01:15.119,0:01:18.560
the fundamental probability postulate
0:01:17.360,0:01:21.520
for measurement
0:01:18.560,0:01:21.520
is as follows
0:01:21.840,0:01:25.280
possible results of a measurement of the
0:01:24.640,0:01:28.799
operator
0:01:25.280,0:01:31.119
A are eigenvalues of A
0:01:28.799,0:01:31.119
only
0:01:32.960,0:01:36.560
after measurement of eigenvalue a_n the
0:01:35.439,0:01:39.360
resulting state
0:01:36.560,0:01:40.000
is the eigenstate |a_n> so something
0:01:39.360,0:01:41.520
mysterious
0:01:40.000,0:01:43.200
happens when we carry out this
0:01:41.520,0:01:44.880
measurement and we'll look into a bit
0:01:43.200,0:01:45.759
more detail about what happens when we
0:01:44.880,0:01:49.280
make measurements
0:01:45.759,0:01:51.040
in another video the probability of
0:01:49.280,0:01:53.520
finding this outcome for state
0:01:51.040,0:01:54.560
psi is the inner product of the
0:01:53.520,0:01:57.600
eigenstate a
0:01:54.560,0:01:58.320
n with psi modulus squared this is what
0:01:57.600,0:02:01.119
we refer to
0:01:58.320,0:02:01.119
as the Born rule
0:02:01.680,0:02:06.159
and finally in the absence of
0:02:03.759,0:02:07.280
measurement states evolve unitarily
0:02:06.159,0:02:09.280
according to the time dependent
0:02:07.280,0:02:12.480
schroedinger equation i h bar d
0:02:09.280,0:02:15.200
psi di by dt is equal to h psi
0:02:12.480,0:02:16.720
of t let's take a look in the worked
0:02:15.200,0:02:19.760
example area at
0:02:16.720,0:02:21.520
what unitarily here means but otherwise
0:02:19.760,0:02:23.840
these are the postulates of quantum
0:02:21.520,0:02:23.840
mechanics
0:02:29.200,0:02:33.840
okay so in general a unitary operator
0:02:32.239,0:02:36.480
takes the following form
0:02:33.840,0:02:37.200
we have that u dagger u is equal to the
0:02:36.480,0:02:38.480
identity
0:02:37.200,0:02:40.400
so these could be matrices and this
0:02:38.480,0:02:42.000
would be the identity matrix these
0:02:40.400,0:02:45.360
could be operators and this would be
0:02:42.000,0:02:46.239
the identity operator so the reason we
0:02:45.360,0:02:48.239
require this
0:02:46.239,0:02:49.360
is that we'd like our states to remain
0:02:48.239,0:02:51.680
normalized
0:02:49.360,0:02:53.120
at all times so remember all physical
0:02:51.680,0:02:55.280
states in quantum mechanics are
0:02:53.120,0:02:58.800
normalized
0:02:55.280,0:03:02.800
so we'd like that if we start with a
0:02:58.800,0:03:04.400
state psi at time naught
0:03:02.800,0:03:06.000
this better be normalized this is a
0:03:04.400,0:03:09.840
better equal one
0:03:06.000,0:03:09.840
but at a later time t
0:03:10.159,0:03:13.920
this had also better remain
0:03:12.000,0:03:15.360
normalized and this is also a better
0:03:13.920,0:03:18.720
equal one
0:03:15.360,0:03:22.640
now we
0:03:18.720,0:03:22.640
for various reasons we're aware that
0:03:23.760,0:03:27.920
oh sorry we should be able to write our
0:03:25.760,0:03:29.599
state psi of t as some kind of
0:03:27.920,0:03:32.000
operator and we don't need to presume
0:03:29.599,0:03:35.040
it's unitary just yet
0:03:32.000,0:03:39.519
which is a function of t and t naught
0:03:35.040,0:03:39.519
acting on our state psi of t naught
0:03:40.799,0:03:45.519
and this condition up here tells us that
0:03:45.680,0:03:49.440
since we have this
0:03:48.640,0:03:51.440
equal to one
0:03:49.440,0:03:53.040
so we'd like the Hermitian conjugate
0:03:51.440,0:03:56.720
of this acting on itself
0:03:53.040,0:03:57.680
to be equal to one and this tells us
0:03:56.720,0:04:01.280
that psi
0:03:57.680,0:04:05.120
of t naught u dagger
0:04:01.280,0:04:08.239
t comma t naught u
0:04:05.120,0:04:12.239
t comma t naught u
0:04:08.239,0:04:15.840
that's right psi of t naught
0:04:12.239,0:04:18.639
must equal one okay so
0:04:15.840,0:04:19.040
then we know that this thing here must
0:04:18.639,0:04:22.160
be
0:04:19.040,0:04:23.440
the identity because if this is the
0:04:22.160,0:04:25.440
identity then we have that
0:04:23.440,0:04:27.199
psi of t naught inner product psi of t
0:04:25.440,0:04:28.400
naught is equal to one but that was this
0:04:27.199,0:04:31.520
condition up here
0:04:28.400,0:04:32.880
so if we start from this condition and
0:04:31.520,0:04:34.000
we require this condition as well the
0:04:32.880,0:04:37.840
only way for that to work
0:04:34.000,0:04:40.240
is if this operator u is unitary
0:04:37.840,0:04:41.919
and in fact we know what it is because
0:04:40.240,0:04:43.600
in quantum mechanics we have the time
0:04:41.919,0:04:45.520
dependent schrodinger equation
0:04:43.600,0:04:48.240
and if you remember back we know that
0:04:45.520,0:04:52.320
psi at time t
0:04:48.240,0:04:56.639
is equal to e to the minus i hamiltonian
0:04:52.320,0:04:59.759
t minus t naught over h bar
0:04:56.639,0:04:59.759
psi of t naught
0:05:01.120,0:05:04.160
which fulfills this condition so that's
0:05:03.520,0:05:06.560
what we mean
0:05:04.160,0:05:07.919
by a unitary operator in fact in
0:05:06.560,0:05:10.160
general
0:05:07.919,0:05:11.280
a unitary operator can always be written
0:05:10.160,0:05:14.160
as
0:05:11.280,0:05:14.880
e to the i hermitian operator and you
0:05:14.160,0:05:16.880
can check that
0:05:14.880,0:05:20.479
quite straightforwardly okay thank you
0:05:16.880,0:05:20.479
for your time
V7.5 Schrödinger's cat demo
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
the measurement problem of quantum mechanics; a demonstration of the Schrödinger's cat thought experiment using a quantum computer; interpretations of quantum mechanics.
0:00:01.920,0:00:05.040
hello
0:00:03.280,0:00:07.200
in this video we're going to talk about
0:00:05.040,0:00:09.440
schrodinger's cat
0:00:07.200,0:00:10.960
and remarkably i'm actually going to be
0:00:09.440,0:00:13.280
able to show you an experimental
0:00:10.960,0:00:15.120
implementation of it but don't worry no
0:00:13.280,0:00:17.039
cat is going to be hurt
0:00:15.120,0:00:18.720
and Geoffrey the dog is going to be fine
0:00:17.039,0:00:21.760
no animals will be involved
0:00:18.720,0:00:23.920
i'll be the cat so
0:00:21.760,0:00:25.039
schrodinger's cat tells us about what's
0:00:23.920,0:00:26.560
called the measurement problem in
0:00:25.039,0:00:28.000
quantum mechanics or it's a particular
0:00:26.560,0:00:31.119
example of it
0:00:28.000,0:00:33.280
so we know that we can take
0:00:31.119,0:00:35.360
a system and we can create a quantum
0:00:33.280,0:00:38.640
superposition of quantum states
0:00:35.360,0:00:39.920
so for example we could take a two level
0:00:38.640,0:00:43.200
system
0:00:39.920,0:00:43.600
we could describe the system by the
0:00:43.200,0:00:46.480
state
0:00:43.600,0:00:48.079
psi which is equal to for example we
0:00:46.480,0:00:50.000
could call it spin up
0:00:48.079,0:00:51.600
and spin down and we could create an
0:00:50.000,0:00:54.399
equally weighted superposition
0:00:51.600,0:00:57.039
of up and down or we could call the two
0:00:54.399,0:00:59.359
states zero and one
0:00:57.039,0:01:00.559
and we could say that we want state
0:01:00.559,0:01:04.559
zero plus one over root two through two
0:01:02.879,0:01:06.400
here's just for normalization
0:01:04.559,0:01:09.680
and this would be an equally weighted
0:01:06.400,0:01:12.560
superposition of those two states
0:01:09.680,0:01:13.439
so spin is a good example we can create
0:01:12.560,0:01:15.119
a
0:01:13.439,0:01:17.280
superposition of spin up and spin down
0:01:15.119,0:01:18.799
quite straightforwardly
0:01:17.280,0:01:20.479
by measuring in a perpendicular
0:01:18.799,0:01:23.600
direction for example
0:01:20.479,0:01:25.759
so we might think that this is
0:01:23.600,0:01:27.600
this quantum superposition which we
0:01:25.759,0:01:29.280
don't see on everyday scales
0:01:27.600,0:01:31.119
maybe it's just some quirk of small
0:01:29.280,0:01:32.240
systems and the fact that we don't see
0:01:31.119,0:01:33.600
it is just you know
0:01:32.240,0:01:35.600
no more weird than the fact that we
0:01:33.600,0:01:36.400
don't experience the world on the scale
0:01:35.600,0:01:39.439
of electrons
0:01:36.400,0:01:42.000
say schrodinger's paradox shows us
0:01:39.439,0:01:43.520
that it's much more of a problem than
0:01:42.000,0:01:46.399
that and actually it is a big
0:01:43.520,0:01:48.079
problem for philosophy and physics
0:01:46.399,0:01:52.240
on everyday scales
0:01:48.079,0:01:53.520
so the basic idea of the essence
0:01:52.240,0:01:56.560
of the paradox was actually come up with
0:01:53.520,0:01:58.719
by Einstein in letters to Schrodinger
0:01:56.560,0:01:59.920
but Schrodinger made one key development
0:01:58.719,0:02:03.439
which is i think why
0:01:59.920,0:02:06.880
it's become such an important
0:02:03.439,0:02:08.560
idea more widely than physics and that
0:02:06.880,0:02:09.280
is that schrodinger changed the story
0:02:08.560,0:02:12.480
into
0:02:09.280,0:02:13.840
involving a cat so
0:02:12.480,0:02:15.200
cats are quite popular and i think this
0:02:13.840,0:02:15.760
has probably helped with the popularity
0:02:15.200,0:02:18.879
of the
0:02:15.760,0:02:21.360
of the paradox so
0:02:18.879,0:02:22.319
here's a retelling of schrodinger's
0:02:21.360,0:02:24.319
paradox
0:02:22.319,0:02:25.599
so you take some measurement device
0:02:24.319,0:02:27.520
which can measure a
0:02:25.599,0:02:28.879
two level system like a spin one half so
0:02:27.520,0:02:31.040
it could be this
0:02:28.879,0:02:33.440
and you create you prepare the state
0:02:31.040,0:02:35.120
in a quantum superposition of two states
0:02:33.440,0:02:38.239
let's say they're equal probability like
0:02:35.120,0:02:40.640
this or you can call them zero and one
0:02:38.239,0:02:41.680
and you set a timer on your measurement
0:02:40.640,0:02:43.760
device so it's going to make its
0:02:41.680,0:02:48.080
measurement after one minute say
0:02:43.760,0:02:48.720
now you rig up the device to a vial of
0:02:48.080,0:02:51.360
poison
0:02:48.720,0:02:51.760
so that if the device measures spin up
0:02:51.760,0:02:56.480
it's going to break the vial open
0:02:54.879,0:02:58.080
and release the poison and if it
0:02:56.480,0:02:58.640
measures spin down it's not going to do
0:02:58.080,0:02:59.920
that
0:02:58.640,0:03:01.519
now you take the whole set up and you
0:02:59.920,0:03:03.599
stick it in a box and then you go and
0:03:01.519,0:03:05.920
get a cat and you put a cat in the box
0:03:03.599,0:03:07.120
and you put a lid on the box now the
0:03:05.920,0:03:08.000
measurement is going to be made after
0:03:07.120,0:03:09.519
one minute
0:03:08.000,0:03:11.200
you wait for two minutes and then you
0:03:09.519,0:03:15.040
take the lid off the box
0:03:11.200,0:03:16.400
and just as usual you well as usual as
0:03:15.040,0:03:19.440
doing this experiment is
0:03:16.400,0:03:20.400
you find either a dead cat and a
0:03:19.440,0:03:22.319
measurement device
0:03:20.400,0:03:23.440
saying that it measured spin up or you
0:03:22.319,0:03:24.799
find it a live cat
0:03:23.440,0:03:26.879
and a measurement device saying they
0:03:24.799,0:03:30.000
should spin down
0:03:26.879,0:03:30.799
okay so that's pretty unethical what
0:03:30.000,0:03:33.200
happened
0:03:30.799,0:03:33.840
before the two minutes were up though
0:03:33.200,0:03:35.440
so
0:03:33.840,0:03:38.000
in the first minute presumably the cat
0:03:35.440,0:03:40.560
is alive and happy enough as happy as a
0:03:38.000,0:03:42.640
cat can be when it's been shut in a box
0:03:40.560,0:03:45.280
but in the second minute the measurement
0:03:42.640,0:03:47.280
device has made its measurement
0:03:45.280,0:03:49.440
and it has a 50 per cent chance of measuring up
0:03:47.280,0:03:51.280
or down and so it has a 50 per cent
0:03:49.440,0:03:53.280
chance of breaking the vial or not and
0:03:51.280,0:03:54.480
the cat has a 50 per cent chance of being dead or
0:03:53.280,0:03:56.080
alive
0:03:54.480,0:03:58.480
now we don't know which of those two
0:03:56.080,0:04:00.640
things it is so in that second minute
0:03:58.480,0:04:01.920
we're still only able to write down a
0:04:00.640,0:04:02.799
quantum state for the cat which
0:04:01.920,0:04:04.799
describes
0:04:02.799,0:04:06.560
the system as an equally weighted
0:04:04.799,0:04:09.200
superposition of the two
0:04:06.560,0:04:09.599
outcomes and so then we need to describe
0:04:09.200,0:04:11.760
the
0:04:09.599,0:04:13.599
state of the cat as an equally weighted
0:04:11.760,0:04:14.000
quantum superposition of both dead and
0:04:13.599,0:04:16.000
alive
0:04:14.000,0:04:18.400
in that second minute and that's
0:04:16.000,0:04:20.000
Schrodinger's paradox
0:04:18.400,0:04:22.240
that's an absurd statement and
0:04:20.000,0:04:23.680
schrodinger made the statement to try
0:04:22.240,0:04:25.040
and show that something was going wrong
0:04:23.680,0:04:27.120
in our logic here
0:04:25.040,0:04:28.240
but these days a lot of people accept
0:04:27.120,0:04:29.680
that that is a
0:04:28.240,0:04:31.600
fact of reality that you can have
0:04:29.680,0:04:32.960
quantum superpositions of large scale
0:04:31.600,0:04:34.110
objects
0:04:32.960,0:04:36.240
now
0:04:36.240,0:04:41.840
i think so it's important to note that
0:04:39.520,0:04:42.720
this isn't the same as taking a coin and
0:04:41.840,0:04:44.479
tossing it
0:04:42.720,0:04:46.400
and catching it and slapping it on your
0:04:44.479,0:04:48.000
hand and saying i don't know whether
0:04:46.400,0:04:49.919
it's heads or tails so i'm going to
0:04:48.000,0:04:52.400
describe that by
0:04:49.919,0:04:54.160
a 50 per cent probability for each because in
0:04:52.400,0:04:56.560
that case it really is heads or tails i
0:04:54.160,0:04:58.000
just don't know which one it is and
0:04:56.560,0:04:59.680
there's a fundamental difference between
0:04:58.000,0:05:01.680
that and
0:04:59.680,0:05:03.520
making an equally weighted quantum
0:05:01.680,0:05:05.440
superposition
0:05:03.520,0:05:07.199
and i think the easiest way to see that
0:05:05.440,0:05:09.039
there's a difference is that we can make
0:05:07.199,0:05:11.120
use of
0:05:09.039,0:05:12.400
quantum superpositions whereas we can't
0:05:11.120,0:05:15.360
make use of the fact that we don't know
0:05:12.400,0:05:16.720
whether the coin is heads or tails
0:05:15.360,0:05:18.800
and an easy way to see that we can make
0:05:16.720,0:05:20.240
use of it is by
0:05:18.800,0:05:22.720
putting it to use in a quantum
0:05:20.240,0:05:25.440
computer so a quantum computer can carry
0:05:22.720,0:05:27.520
out certain calculations exponentially
0:05:25.440,0:05:28.400
faster than any classical computer ever
0:05:27.520,0:05:31.840
could
0:05:28.400,0:05:33.600
okay so they make use of two
0:05:31.840,0:05:35.199
key properties of quantum mechanics one
0:05:33.600,0:05:36.320
is superpositions
0:05:35.199,0:05:38.720
and the other is what's called quantum
0:05:36.320,0:05:39.280
entanglement so in the present setting
0:05:38.720,0:05:41.919
actually
0:05:39.280,0:05:43.120
entanglement isn't that mysterious it
0:05:41.919,0:05:45.280
was actually come up with
0:05:43.120,0:05:46.160
by schroedinger in the phrases come up
0:05:45.280,0:05:48.479
with in German
0:05:46.160,0:05:51.360
between in these letters to Einstein
0:05:48.479,0:05:52.880
regarding Schrodinger's cat
0:05:51.360,0:05:54.479
and it's actually in this case it's
0:05:52.880,0:05:56.160
just a simple statement of conditional
0:05:54.479,0:05:58.080
probabilities effectively
0:05:56.160,0:05:59.440
so if we think what's happening in that
0:05:58.080,0:06:02.800
box the
0:05:59.440,0:06:04.479
spin is in a superposition of
0:06:02.800,0:06:05.759
being a spin up and spin down
0:06:04.479,0:06:07.280
now the measurement device after it's
0:06:05.759,0:06:08.160
made its measurement but before we've
0:06:07.280,0:06:10.639
looked
0:06:08.160,0:06:11.440
either measures spin up or spin down
0:06:10.639,0:06:13.199
but of course
0:06:11.440,0:06:14.960
what it measures is
0:06:13.199,0:06:17.759
conditional on what the state is
0:06:14.960,0:06:18.639
so if the measurement device finds
0:06:17.759,0:06:20.560
spin up
0:06:18.639,0:06:22.560
that means that the spin must be spin up
0:06:20.560,0:06:26.560
okay it's conditional on that
0:06:22.560,0:06:28.160
if the spin is measured to be spin up
0:06:26.560,0:06:30.000
so if the spin is spin up then it's
0:06:28.160,0:06:32.160
measured to be spin up if it's spin down
0:06:30.000,0:06:33.919
it's measured to be spin down
0:06:32.160,0:06:36.240
okay so it's a form of conditional
0:06:33.919,0:06:38.000
probability but in quantum mechanics
0:06:36.240,0:06:39.440
of course we use amplitudes rather than
0:06:38.000,0:06:42.319
probabilities and then we use the Born
0:06:39.440,0:06:43.680
rule to get the actual probabilities
0:06:42.319,0:06:45.759
so in this case entanglement is no
0:06:43.680,0:06:47.759
weirder than saying that if the device
0:06:45.759,0:06:49.360
has measured spin up that means that it
0:06:47.759,0:06:50.639
must be the case that the spin is spin
0:06:49.360,0:06:53.759
up
0:06:50.639,0:06:55.280
and so on and so that's a form of
0:06:53.759,0:06:56.960
entanglement which
0:06:55.280,0:06:58.560
schrodinger says that the measurement
0:06:56.960,0:07:00.240
device has become entangled with the
0:06:58.560,0:07:01.280
quantum system and that's how we've kind
0:07:00.240,0:07:03.440
of scaled up this
0:07:01.280,0:07:04.639
this superposition to a large scale
0:07:03.440,0:07:06.319
thing like the cat
0:07:04.639,0:07:08.400
because then if the measurement device
0:07:06.319,0:07:11.039
measures spin up it breaks the vial
0:07:08.400,0:07:11.840
it kills the cat and so then
0:07:11.840,0:07:14.880
you've entangled the state of the cat
0:07:13.520,0:07:15.360
with the state of the measurement device
0:07:14.880,0:07:21.120
and
0:07:15.360,0:07:23.599
with the state of the spin-half particle
0:07:21.120,0:07:25.840
so the fact that we can use quantum
0:07:23.599,0:07:27.280
computers to do this kind of thing now
0:07:25.840,0:07:28.880
means that i can do this experiment in
0:07:27.280,0:07:30.880
my house
0:07:28.880,0:07:32.080
and the reason for this is that various
0:07:30.880,0:07:34.319
companies which
0:07:32.080,0:07:36.479
have built quantum computers have
0:07:34.319,0:07:38.880
provided free access to them online
0:07:36.479,0:07:40.319
and so you can just sign up for an
0:07:38.880,0:07:41.440
account and you can just use a quantum
0:07:40.319,0:07:43.039
computer now
0:07:41.440,0:07:45.599
you can only use a few qubits at the
0:07:43.039,0:07:48.639
moment so a qubit is a quantum bit:
0:07:45.599,0:07:50.160
a bit of information would be
0:07:48.639,0:07:52.720
either zero or one
0:07:50.160,0:07:53.919
hence the nomenclature zero and one
0:07:52.720,0:07:56.160
here
0:07:53.919,0:07:57.520
a qubit a quantum bit is also either
0:07:56.160,0:07:59.360
zero or one but it can
0:07:57.520,0:08:00.639
also be a quantum superposition of zero
0:07:59.360,0:08:01.919
and one
0:08:00.639,0:08:03.759
for example this equally weighted
0:08:01.919,0:08:04.879
superposition we just considered but it
0:08:03.759,0:08:09.680
can be any other
0:08:04.879,0:08:12.080
superposition so
0:08:09.680,0:08:13.840
in quantum computing there's you can
0:08:12.080,0:08:17.440
write quantum algorithms
0:08:13.840,0:08:20.879
mainly for testing your
0:08:17.440,0:08:23.039
approaches to quantum computing
0:08:20.879,0:08:24.639
but we can use one I can log into one
0:08:23.039,0:08:25.599
now so i'm going to do that in a
0:08:24.639,0:08:28.160
second i'll use
0:08:25.599,0:08:28.960
one of IBM's quantum computers in New
0:08:28.160,0:08:31.599
York
0:08:28.960,0:08:33.440
and what we can do is create this
0:08:31.599,0:08:35.440
superposition and then we can measure it
0:08:33.440,0:08:36.000
and say are you in state 0 or in state
0:08:35.440,0:08:37.760
1.
0:08:36.000,0:08:40.560
and there will be a 50 per cent probability for
0:08:37.760,0:08:43.120
it giving either of the outcomes
0:08:40.560,0:08:45.440
okay so let me just show you the setup
0:08:43.120,0:08:48.800
i'll switch over to
0:08:45.440,0:08:51.440
showing you my screen so this is
0:08:48.800,0:08:53.279
the ibm quantum experience and i'm
0:08:51.440,0:08:56.320
logged into
0:08:53.279,0:08:59.120
the quantum computer in new york and
0:08:56.320,0:09:00.480
here is so you can see here's the code
0:08:59.120,0:09:02.000
right there's a
0:09:00.480,0:09:03.519
python interface for it it's pretty
0:09:02.000,0:09:06.399
straightforward
0:09:03.519,0:09:08.240
but actually the way you write quantum
0:09:06.399,0:09:11.760
algorithms tends to be in this much
0:09:08.240,0:09:14.240
simpler method using diagrams
0:09:11.760,0:09:16.000
so you have various tracks running along
0:09:14.240,0:09:17.600
where you initialize qubits
0:09:16.000,0:09:18.800
and then you perform different
0:09:17.600,0:09:20.160
operations on them -- what are called
0:09:18.800,0:09:21.839
quantum gates --
0:09:20.160,0:09:23.360
and they can transform qubits and they
0:09:21.839,0:09:25.519
can entangle them disentangle and this
0:09:23.360,0:09:28.720
kind of thing
0:09:25.519,0:09:31.040
so all i want to do is create this
0:09:28.720,0:09:31.839
superposition i just showed you and
0:09:31.040,0:09:34.160
so
0:09:31.839,0:09:36.000
the convention is that
0:09:34.160,0:09:38.480
you'll you start off with a load of
0:09:36.000,0:09:40.240
qubits prepared in the state zero so
0:09:38.480,0:09:41.680
this could be spin down for example if
0:09:40.240,0:09:42.480
they were using spins actually i think
0:09:41.680,0:09:44.720
they're using
0:09:42.480,0:09:45.519
the polarizations of light in these
0:09:45.519,0:09:50.800
experimental setups
0:09:48.560,0:09:52.560
and so we need to act some kind of
0:09:50.800,0:09:54.160
operation that turns our
0:09:52.560,0:09:56.000
state zero into an equally weighted
0:09:54.160,0:09:56.880
superposition of state zero plus state
0:09:56.000,0:09:58.880
one
0:09:56.880,0:10:00.800
over root two now the operation that
0:09:58.880,0:10:02.720
does that is given the symbol H
0:10:00.800,0:10:04.000
it's what's called a Hadamard gate so
0:10:02.720,0:10:05.680
it's not a Hamiltonian
0:10:04.000,0:10:08.079
it's just someone else's name began
0:10:05.680,0:10:09.920
with H so Hadamard will switch our state
0:10:08.079,0:10:11.279
0 to an equally weighted superposition
0:10:09.920,0:10:13.839
of 0 and 1.
0:10:11.279,0:10:15.040
and then this symbol here is a
0:10:13.839,0:10:16.720
measurement symbol
0:10:15.040,0:10:18.800
and what it does is it takes our quantum
0:10:16.720,0:10:20.240
bit our qubit and it's going to project
0:10:18.800,0:10:21.600
it down into a classical bit
0:10:20.240,0:10:24.079
because once the measurement's been made
0:10:21.600,0:10:26.079
we've got a classical outcome read out
0:10:24.079,0:10:28.959
so it's going to measure the state and
0:10:26.079,0:10:28.959
and give a readout
0:10:29.920,0:10:35.760
so you can set it running
0:10:32.959,0:10:37.600
i can go to jobs over here i've actually
0:10:35.760,0:10:39.200
run it already but i can rerun it for
0:10:37.600,0:10:41.040
you you can run it many times
0:10:39.200,0:10:42.560
and and that's usually the use of this
0:10:41.040,0:10:43.440
but i'm just going to run it once just
0:10:42.560,0:10:47.120
because i want to
0:10:43.440,0:10:49.760
get one outcome or the other now
0:10:47.120,0:10:50.640
to be schrodinger's cat what i want
0:10:49.760,0:10:52.240
to do is
0:10:50.640,0:10:53.920
i'm going to do something different
0:10:52.240,0:10:54.800
depending on the outcome of this quantum
0:10:53.920,0:10:58.160
measurement
0:10:54.800,0:10:59.120
so let's say if we measure zero i'll
0:10:58.160,0:11:02.800
go and sit
0:10:59.120,0:11:04.640
on the left side of my sofa
0:11:02.800,0:11:06.480
and or actually I'll sit on the
0:11:04.640,0:11:08.880
right side of the sofa over there
0:11:06.480,0:11:10.800
and if it measures one i'll go and sit
0:11:08.880,0:11:14.000
on the left side of the sofa
0:11:10.800,0:11:17.200
okay so we can set our
0:11:14.000,0:11:19.440
job going so we set
0:11:17.200,0:11:19.440
up
0:11:20.399,0:11:24.640
logged into there so let's run it
0:11:25.839,0:11:29.839
okay so it's sent off to new york and
0:11:27.920,0:11:32.079
we're queued and we can expect to wait
0:11:29.839,0:11:36.800
about an hour for that to go through
0:11:32.079,0:11:36.800
so i'll fast forward on the video
0:11:39.440,0:11:44.320
so the quantum computer measured the
0:11:42.399,0:11:45.839
state to be in state zero
0:11:44.320,0:11:47.440
and so i came and sat over here on the
0:11:45.839,0:11:48.720
right of the sofa
0:11:47.440,0:11:50.399
if it had measured one i would have been
0:11:48.720,0:11:52.240
sat over there where Geoffrey is on the
0:11:50.399,0:11:54.959
left
0:11:52.240,0:11:56.959
so after the computer made its
0:11:54.959,0:11:58.000
measurement and after i'd looked at that
0:11:56.959,0:12:00.000
measurement
0:11:58.000,0:12:01.519
but before you'd look to see which of
0:12:00.000,0:12:03.200
the two things i did
0:12:01.519,0:12:05.440
then you had to assign an equally
0:12:03.200,0:12:06.240
weighted probability to me being here or
0:12:05.440,0:12:07.760
there
0:12:06.240,0:12:09.839
and since you're ultimately describing a
0:12:07.760,0:12:10.959
quantum state you actually had to assign
0:12:09.839,0:12:13.279
an equally weighted
0:12:10.959,0:12:16.560
quantum superposition of amplitudes of
0:12:13.279,0:12:16.560
me being here and me being there
0:12:16.880,0:12:22.079
so how we
0:12:19.920,0:12:24.880
explain this situation is a matter of
0:12:22.079,0:12:27.360
philosophical interpretation
0:12:24.880,0:12:28.240
we describe the different possible
0:12:27.360,0:12:30.639
explanations
0:12:28.240,0:12:31.360
as interpretations of quantum mechanics
0:12:30.639,0:12:33.200
rather than
0:12:31.360,0:12:34.800
theories of quantum mechanics because
0:12:33.200,0:12:36.160
they're not strictly theories in the
0:12:34.800,0:12:39.040
scientific sense
0:12:36.160,0:12:40.480
since they all make the same testable
0:12:39.040,0:12:41.680
predictions they don't make different
0:12:40.480,0:12:43.279
testable predictions
0:12:41.680,0:12:44.959
because to do so they'd have to disagree
0:12:43.279,0:12:45.680
with the mathematics of quantum
0:12:44.959,0:12:47.760
mechanics
0:12:45.680,0:12:49.519
which are extremely well tested against
0:12:47.760,0:12:52.560
reality
0:12:49.519,0:12:54.320
there are some cases where theories
0:12:52.560,0:12:55.680
truly diverge from the predictions of
0:12:54.320,0:12:56.880
quantum mechanics
0:12:55.680,0:12:59.600
and we'll take a look at some of those
0:12:56.880,0:13:00.399
in a second so to look at some possible
0:12:59.600,0:13:07.839
options let's
0:13:00.399,0:13:07.839
return to the worked example area
0:13:10.480,0:13:14.560
so probably the most mainstream
0:13:12.639,0:13:15.839
interpretation of quantum mechanics
0:13:14.560,0:13:17.920
is what's called the copenhagen
0:13:15.839,0:13:18.639
interpretation developed in copenhagen
0:13:17.920,0:13:20.320
by
0:13:18.639,0:13:22.160
people such as Niels Bohr and
0:13:20.320,0:13:24.720
Werner Heisenberg so
0:13:22.160,0:13:25.440
in the copenhagen interpretation we say
0:13:24.720,0:13:29.040
that
0:13:25.440,0:13:30.320
something special happens
0:13:29.040,0:13:31.920
when a measurement is made
0:13:30.320,0:13:33.600
and that special thing is what's called
0:13:31.920,0:13:36.959
the collapse of the wave function
0:13:33.600,0:13:39.920
or wave function collapse so when we go
0:13:36.959,0:13:41.440
from the quantum superposition 0 plus 1
0:13:39.920,0:13:44.399
over root 2.
0:13:41.440,0:13:49.680
let's write that down so we start in
0:13:44.399,0:13:52.639
this state
0:13:49.680,0:13:52.639
measurement occurs
0:13:55.040,0:13:58.480
and in our case we found the state
0:13:57.680,0:13:59.839
zero
0:13:58.480,0:14:01.680
so the state of the system really has
0:13:59.839,0:14:03.920
changed
0:14:01.680,0:14:05.440
but this leads to a whole set of
0:14:03.920,0:14:07.360
philosophical problems
0:14:05.440,0:14:09.760
for example what constitutes a
0:14:07.360,0:14:13.040
measurement? In the case of the cat
0:14:09.760,0:14:16.000
when i've looked into the box to see
0:14:13.040,0:14:16.720
the cat i find it either dead or alive
0:14:16.000,0:14:18.399
so then
0:14:16.720,0:14:19.760
that could constitute the measurement my
0:14:18.399,0:14:21.519
looking but
0:14:19.760,0:14:23.120
why can't the cat collapse the wave
0:14:21.519,0:14:25.120
function?
0:14:23.120,0:14:27.040
Surely the cat which either dies or does
0:14:25.120,0:14:27.839
not is is a pretty good measurement
0:14:27.040,0:14:29.680
device
0:14:27.839,0:14:31.120
but that would mean that the wave
0:14:29.680,0:14:32.160
function had collapsed before i opened
0:14:31.120,0:14:33.760
the box
0:14:32.160,0:14:35.279
but similarly why can't the measurement
0:14:33.760,0:14:36.320
device itself count as the measurement
0:14:35.279,0:14:38.800
device
0:14:36.320,0:14:41.519
and so the wave function is collapsed
0:14:38.800,0:14:43.360
before it gets to the cat
0:14:41.519,0:14:45.199
at what point something constitutes a
0:14:43.360,0:14:47.120
measurement does
0:14:45.199,0:14:49.120
is it the size of the measurement device
0:14:47.120,0:14:52.320
is it like the number of particles in it?
0:14:49.120,0:14:55.519
Is it its physical size?
0:14:52.320,0:14:56.160
The Copenhagen interpretation doesn't
0:14:55.519,0:14:59.519
really
0:14:56.160,0:15:00.880
comment on any of these things a
0:14:59.519,0:15:02.720
view that often goes along with the
0:15:00.880,0:15:05.760
copenhagen interpretation although
0:15:02.720,0:15:07.680
isn't necessarily a part of it is
0:15:05.760,0:15:09.279
encapsulated in the maxim 'shut up and
0:15:07.680,0:15:10.399
calculate' which says:
0:15:09.279,0:15:12.959
don't worry about these kinds of
0:15:10.399,0:15:14.240
questions just use the maths and it'll
0:15:12.959,0:15:15.680
make predictions which are extremely
0:15:14.240,0:15:16.720
well tested that's all you need to worry
0:15:15.680,0:15:18.399
about
0:15:16.720,0:15:20.079
actually because it's basically the
0:15:18.399,0:15:21.360
oldest interpretation or the oldest
0:15:20.079,0:15:23.199
attempt to interpret quantum
0:15:21.360,0:15:26.320
mechanics
0:15:23.199,0:15:27.600
the
0:15:26.320,0:15:28.560
interpretation hasn't worried too much
0:15:27.600,0:15:30.560
about trying to
0:15:28.560,0:15:32.560
answer all possible questions; later
0:15:30.560,0:15:36.399
interpretations that came around
0:15:32.560,0:15:36.720
were developed in distinction to
0:15:36.399,0:15:38.639
the
0:15:36.720,0:15:40.079
original copenhagen interpretation and
0:15:38.639,0:15:42.320
they were often addressing
0:15:40.079,0:15:43.680
some particular point which it hadn't
0:15:42.320,0:15:46.959
commented on
0:15:43.680,0:15:49.519
so a another very mainstream
0:15:46.959,0:15:51.120
view is the many worlds interpretation
0:15:49.519,0:15:53.440
so this says that wavefunction collapse
0:15:51.120,0:15:56.560
doesn't occur
0:15:53.440,0:15:58.959
instead it says what happens is that
0:15:56.560,0:16:00.320
when the measurement is carried out by
0:15:58.959,0:16:02.800
the measurement device
0:16:00.320,0:16:03.440
the measurement device entangles itself
0:16:02.800,0:16:06.480
with
0:16:03.440,0:16:09.600
the quantum state so
0:16:06.480,0:16:09.920
remember in schrodinger's description of
0:16:09.600,0:16:12.160
the
0:16:09.920,0:16:13.839
system it says that the conditional
0:16:12.160,0:16:16.000
probability which in this case is a kind
0:16:13.839,0:16:19.040
of conditional amplitude
0:16:16.000,0:16:20.880
for measuring spin up say that's
0:16:19.040,0:16:21.440
conditional on the state actually being
0:16:20.880,0:16:23.519
state
0:16:21.440,0:16:26.000
spin up so we can do it schematically
0:16:23.519,0:16:28.160
something like this
0:16:26.000,0:16:29.680
we can say we start off in this state as
0:16:28.160,0:16:33.920
before: zero
0:16:29.680,0:16:35.600
plus one over root two
0:16:33.920,0:16:37.600
but we have a measurement device there
0:16:35.600,0:16:40.000
which just hasn't made a measurement yet
0:16:37.600,0:16:41.360
so let's write a kind of special
0:16:40.000,0:16:42.959
multiply sign actually this is what's
0:16:41.360,0:16:45.199
called a tensor product
0:16:46.079,0:16:49.120
and there's a measurement
0:16:49.680,0:16:52.240
device
0:16:55.680,0:16:58.959
and what happens when the measurement is
0:16:57.600,0:17:00.880
made is that
0:16:58.959,0:17:02.800
this state now becomes a form of
0:17:00.880,0:17:04.240
conditional probability
0:17:02.800,0:17:06.640
and really conditional amplitude of the
0:17:04.240,0:17:10.720
following form so
0:17:06.640,0:17:13.760
either we have the state 0 and
0:17:10.720,0:17:13.760
we measure zero
0:17:16.240,0:17:22.160
or we have state one and
0:17:19.280,0:17:22.160
we measure one
0:17:33.679,0:17:37.120
we have what's called
0:17:35.679,0:17:38.799
a product state you can think of a
0:17:37.120,0:17:39.600
tensor product much like the usual
0:17:38.799,0:17:41.039
product
0:17:39.600,0:17:42.960
it's just that these things are states
0:17:41.039,0:17:44.840
so they're vectors in the complex
0:17:42.960,0:17:47.840
hilbert space
0:17:44.840,0:17:50.240
and so this is what's called a
0:17:47.840,0:17:51.919
product state and is not entangled
0:17:50.240,0:17:53.679
this cannot be written as a product
0:17:51.919,0:17:55.280
state and anything which can't be
0:17:53.679,0:17:56.320
written as a product state any sum of
0:17:55.280,0:17:57.520
states which can't be written as a
0:17:56.320,0:17:58.720
product state is what's called an
0:17:57.520,0:18:00.880
entangled state
0:17:58.720,0:18:02.640
and so this is a way of mathematically
0:18:00.880,0:18:04.320
codifying what Schroedinger had said
0:18:02.640,0:18:06.640
that the measurement device becomes
0:18:04.320,0:18:08.960
entangled with the state of the system
0:18:06.640,0:18:10.880
and so in the many worlds interpretation
0:18:08.960,0:18:14.400
this is taken quite seriously
0:18:10.880,0:18:16.720
and then some mechanism is
0:18:14.400,0:18:18.799
said to be an operation which causes
0:18:16.720,0:18:20.480
these two possible outcomes to
0:18:18.799,0:18:22.000
separate somehow and not talk to each
0:18:20.480,0:18:25.440
other
0:18:22.000,0:18:26.480
because ultimately you end up so measure
0:18:25.440,0:18:29.120
zero is really
0:18:26.480,0:18:30.480
it ends up with you having seen the
0:18:29.120,0:18:31.200
outcome of measurement zero which in
0:18:30.480,0:18:34.320
this case was a
0:18:31.200,0:18:34.880
dead cat and measure one ends up with
0:18:34.320,0:18:37.200
you
0:18:34.880,0:18:38.240
seeing a live cat and since we don't
0:18:37.200,0:18:40.080
seem to be in
0:18:38.240,0:18:41.360
superpositions of seeing dead and
0:18:40.080,0:18:42.960
alive cats
0:18:41.360,0:18:44.960
we must either be in this state or this
0:18:42.960,0:18:46.000
state and so the idea is that something
0:18:44.960,0:18:47.520
causes these two
0:18:46.000,0:18:49.039
possible outcomes to stop talking to
0:18:47.520,0:18:51.840
each other at some point
0:18:49.039,0:18:53.600
so a common way to explain this is by
0:18:51.840,0:18:56.880
what's called decoherence
0:18:53.600,0:18:58.880
we say that while the
0:18:56.880,0:19:00.320
quantum particle can be in these quantum
0:18:58.880,0:19:02.400
superpositions and so on
0:19:00.320,0:19:04.240
once you start coupling it to big things
0:19:02.400,0:19:05.760
the quantum information starts leaking
0:19:04.240,0:19:07.840
out into the environment
0:19:05.760,0:19:10.480
and while the state is truly still
0:19:07.840,0:19:12.799
behaving in its quantum manner
0:19:10.480,0:19:13.840
it may appear to be classical to any
0:19:12.799,0:19:16.640
given observer
0:19:13.840,0:19:18.559
because the the quantum properties
0:19:16.640,0:19:21.840
have effectively leaked out with the
0:19:18.559,0:19:23.120
leaking of information it's a little bit
0:19:21.840,0:19:24.559
like saying
0:19:23.120,0:19:26.240
if somebody goes and shouts into the
0:19:24.559,0:19:26.799
woods and then you go there five minutes
0:19:26.240,0:19:28.080
later
0:19:26.799,0:19:29.840
we don't know what they shouted you
0:19:28.080,0:19:31.200
weren't there for the shout; in principle
0:19:29.840,0:19:31.600
though you could go around to measure
0:19:31.200,0:19:33.120
all the
0:19:31.600,0:19:35.120
different vibrations of the leaves and
0:19:33.120,0:19:36.400
so on and piece back together what
0:19:35.120,0:19:37.600
the word was they shouted
0:19:36.400,0:19:39.039
but of course that's never going to
0:19:37.600,0:19:40.720
actually happen even though the
0:19:39.039,0:19:41.840
information might technically be there
0:19:40.720,0:19:44.799
in the sense that
0:19:41.840,0:19:45.120
the information is never really lost
0:19:45.120,0:19:48.559
it's useless to try and piece it back
0:19:47.520,0:19:50.559
together
0:19:48.559,0:19:52.000
so quantum information can simply kind
0:19:50.559,0:19:54.480
of seemingly dissipate
0:19:52.000,0:19:56.000
to observers while actually being
0:19:54.480,0:19:58.880
conserved
0:19:56.000,0:20:00.799
so the name many worlds comes from a
0:19:58.880,0:20:03.679
common interpretation of this that
0:20:00.799,0:20:04.480
this branch becomes effectively a
0:20:03.679,0:20:08.320
separate
0:20:04.480,0:20:09.520
universe to this branch which then
0:20:08.320,0:20:10.640
you have the sort of philosophical
0:20:09.520,0:20:12.240
baggage of explaining whether those
0:20:10.640,0:20:14.880
universes coexist
0:20:12.240,0:20:17.360
the idea would be not that they
0:20:14.880,0:20:18.880
physically separate but they're
0:20:17.360,0:20:20.640
sort of in the same place but not
0:20:18.880,0:20:23.360
talking to each other so may as well be
0:20:20.640,0:20:23.360
in different places
0:20:23.840,0:20:27.679
another interpretation which actually is
0:20:25.840,0:20:28.880
a scientific theory is the set of
0:20:27.679,0:20:31.280
objective collapse
0:20:28.880,0:20:32.400
theories so these make
0:20:31.280,0:20:33.280
different predictions to quantum
0:20:32.400,0:20:35.120
mechanics
0:20:33.280,0:20:36.640
but they do so in such a way as to not
0:20:35.120,0:20:38.320
disagree with any measurements which
0:20:36.640,0:20:40.000
have so far been carried out
0:20:38.320,0:20:41.919
so in particular they'll say that
0:20:40.000,0:20:43.360
something does cause the collapse to
0:20:41.919,0:20:45.120
physically happen
0:20:43.360,0:20:46.480
it doesn't have anything to do with
0:20:45.120,0:20:48.320
the person measuring it
0:20:46.480,0:20:49.919
it's just a matter of coupling to a
0:20:48.320,0:20:52.000
large device
0:20:49.919,0:20:53.120
large could either mean physically large
0:20:52.000,0:20:54.720
perhaps
0:20:53.120,0:20:56.480
having different effects of gravity on
0:20:54.720,0:20:58.559
the different ends or it could mean
0:20:56.480,0:21:00.320
large in the sense of lots of particles
0:20:58.559,0:21:01.440
in it
0:21:00.320,0:21:03.760
so there are various theories along
0:21:01.440,0:21:04.240
those lines and they'll tend to say
0:21:03.760,0:21:07.600
that
0:21:04.240,0:21:08.799
the cut-off scale is somewhere
0:21:07.600,0:21:11.200
between the
0:21:08.799,0:21:12.880
quantum scale and our everyday scale
0:21:11.200,0:21:16.000
and in fact we've narrowed it down to
0:21:12.880,0:21:19.440
somewhere in the mesoscopic scale
0:21:16.000,0:21:21.039
so as we are able to maintain quantum
0:21:19.440,0:21:21.840
superpositions on larger and larger
0:21:21.039,0:21:23.600
systems
0:21:21.840,0:21:25.039
we kind of narrow the region in which
0:21:23.600,0:21:26.080
these objective collapse theories could
0:21:25.039,0:21:28.240
operate
0:21:26.080,0:21:30.559
and in principle we'll be able to test
0:21:28.240,0:21:32.000
that full range of theories if we get
0:21:30.559,0:21:33.120
good enough at our experiments
0:21:32.000,0:21:35.200
okay so there are some different
0:21:33.120,0:21:35.919
interpretations of the measurement
0:21:35.200,0:21:38.799
problem
0:21:35.919,0:21:41.440
and as shown clearly by the
0:21:38.799,0:21:43.679
Schrodinger's cat paradox
0:21:41.440,0:21:44.799
and it's interesting to try and think
0:21:43.679,0:21:47.200
which you agree with
0:21:44.799,0:21:48.240
perhaps you don't agree with any
0:21:47.200,0:21:49.200
or perhaps you have one that you
0:21:48.240,0:21:53.200
particularly like
0:21:49.200,0:21:53.200
okay thank you for your time
V8.1 The quantum harmonic oscillator
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
solution of the quantum harmonic oscillator by reduction to Hermite's equation; orthogonality and completeness of Hermite polynomials.
0:00:00.080,0:00:04.080
hello in this video we're going to take
0:00:02.080,0:00:06.080
a look at the quantum harmonic
0:00:04.080,0:00:08.160
oscillator one of the most important
0:00:06.080,0:00:10.080
problems in quantum mechanics so here's
0:00:08.160,0:00:11.920
the potential
0:00:10.080,0:00:13.679
written now in operator form as we're
0:00:11.920,0:00:15.679
very familiar with that
0:00:13.679,0:00:17.600
but let's go back and write out the time
0:00:15.679,0:00:20.720
independent schrodinger equation
0:00:17.600,0:00:22.480
in terms of wave function so we have our
0:00:20.720,0:00:24.880
kinetic energy operator
0:00:22.480,0:00:27.519
and our potential energy here and we see
0:00:24.880,0:00:30.560
that it's just a quadratic
0:00:27.519,0:00:31.439
potential and we'll have an infinite
0:00:30.560,0:00:34.480
tower of
0:00:31.439,0:00:37.040
eigenfunctions labeled by integers n
0:00:34.480,0:00:38.399
and a set of eigen energies associated
0:00:37.040,0:00:39.840
with them so they're bound states and
0:00:38.399,0:00:42.840
there's an infinite number of them
0:00:39.840,0:00:44.559
the potential looks like this a
0:00:42.840,0:00:47.520
quadratic and
0:00:44.559,0:00:48.960
drawing the states in fact
0:00:47.520,0:00:50.800
we can guess roughly what they look like
0:00:48.960,0:00:52.480
again from thinking about the infinite
0:00:50.800,0:00:53.039
potential well and the finite potential
0:00:52.480,0:00:54.800
well
0:00:53.039,0:00:58.079
and in fact the results that we're
0:00:54.800,0:01:00.719
going to find look something like this
0:00:58.079,0:01:01.359
so we have a ground state wave
0:01:00.719,0:01:03.600
function
0:01:01.359,0:01:05.280
and it has a ground state energy
0:01:03.600,0:01:06.960
we're plotting again something like the
0:01:05.280,0:01:08.560
real part it's a particular instant in
0:01:06.960,0:01:10.640
time
0:01:08.560,0:01:12.560
one of the most important features of
0:01:10.640,0:01:14.320
the harmonic oscillator solution
0:01:12.560,0:01:16.400
is that the energy levels are evenly
0:01:14.320,0:01:18.400
spaced i mean i've not drawn them very well
0:01:16.400,0:01:20.320
but the entire infinite ladder of energy
0:01:18.400,0:01:22.240
levels is spaced evenly and that's why
0:01:20.320,0:01:24.400
it turns out to be so useful for all
0:01:22.240,0:01:27.200
other cases
0:01:24.400,0:01:27.759
also in general it's fairly similar
0:01:27.200,0:01:29.680
to why
0:01:27.759,0:01:31.759
taylor series are so important to
0:01:29.680,0:01:32.479
functions while we may not be able to
0:01:31.759,0:01:35.680
solve
0:01:32.479,0:01:36.720
every set of possible equations we
0:01:35.680,0:01:39.840
might come across
0:01:36.720,0:01:42.000
it's often possible to pay attention to
0:01:39.840,0:01:43.200
points near minima and maxima
0:01:42.000,0:01:44.640
those are often the points we're
0:01:43.200,0:01:45.920
interested in, in a
0:01:44.640,0:01:48.000
particular problem especially when
0:01:45.920,0:01:49.920
there's a potential involved
0:01:48.000,0:01:51.600
and around a maximum or a minimum we can
0:01:49.920,0:01:53.680
always expand the potential
0:01:51.600,0:01:54.799
as the first order term being a
0:01:53.680,0:01:56.799
quadratic
0:01:54.799,0:01:58.640
so the harmonic oscillator is
0:01:56.799,0:02:01.200
often a very good approximation
0:01:58.640,0:02:02.399
to questions of physical interest but
0:02:01.200,0:02:05.840
we can also solve it
0:02:02.399,0:02:07.360
exactly and analytically so it'll prove
0:02:05.840,0:02:08.640
useful in a lot of other problems that
0:02:07.360,0:02:11.680
you'll see later on
0:02:08.640,0:02:13.599
in your career so let's take a look
0:02:11.680,0:02:15.120
in this video at solving it the
0:02:13.599,0:02:16.720
old-fashioned way
0:02:15.120,0:02:18.080
by using differential equations then
0:02:16.720,0:02:21.360
we'll look at a much more elegant way
0:02:18.080,0:02:23.680
using operators in a future video
0:02:21.360,0:02:24.400
so first let's change variables slightly
0:02:23.680,0:02:27.280
i will define
0:02:24.400,0:02:28.400
alpha y = x substituting that into
0:02:27.280,0:02:31.120
the time independent schrodinger
0:02:28.400,0:02:34.720
equation we find this
0:02:31.120,0:02:37.040
and if we choose the following
0:02:34.720,0:02:39.120
alpha squared is h bar over m omega we
0:02:37.040,0:02:41.280
can rewrite the equation in the simpler
0:02:39.120,0:02:44.319
form
0:02:41.280,0:02:46.239
where double prime here indicates
0:02:44.319,0:02:48.319
partial derivatives with respect to y
0:02:46.239,0:02:50.560
rather than x as it usually represents
0:02:48.319,0:02:53.120
so this is with respect to y and epsilon
0:02:50.560,0:02:56.160
n here is defined to be
0:02:53.120,0:02:59.120
just a scaling of of the energies E_n
0:02:56.160,0:03:01.120
okay so this is a fairly simple
0:02:59.120,0:03:03.120
looking differential equation
0:03:01.120,0:03:04.159
an ordinary differential equation now
0:03:03.120,0:03:07.200
as y is the only
0:03:04.159,0:03:09.680
variable in the problem it's convenient
0:03:07.200,0:03:12.800
to make the following substitution
0:03:09.680,0:03:15.840
so we'll switch from phi_n(y) to H_n(y)
0:03:12.800,0:03:16.959
multiplied by e to the minus y
0:03:15.840,0:03:19.360
squared over two
0:03:16.959,0:03:20.720
so multiplying the gaussian term when
0:03:19.360,0:03:21.760
we make the substitution we find the
0:03:20.720,0:03:25.040
equation here
0:03:21.760,0:03:27.840
reduces to the following form which
0:03:25.040,0:03:28.799
while not particularly much nicer than
0:03:27.840,0:03:31.040
this one
0:03:28.799,0:03:32.400
was already known so this was known
0:03:31.040,0:03:35.440
before quantum mechanics
0:03:32.400,0:03:36.959
and it's called Hermite's equation where
0:03:35.440,0:03:40.159
i'm certain i'm pronouncing
0:03:36.959,0:03:43.280
this person's name incorrectly
0:03:40.159,0:03:46.080
so it was known already. What
0:03:43.280,0:03:47.840
you do when you're trying to solve
0:03:46.080,0:03:50.239
differential equations analytically is
0:03:47.840,0:03:51.840
simply massage them into a form where
0:03:50.239,0:03:52.799
someone else has solved it hundreds of
0:03:51.840,0:03:54.560
years ago
0:03:52.799,0:03:55.840
and this is no different so the
0:03:54.560,0:03:58.239
solutions H_n(y)
0:03:55.840,0:03:59.760
are what are called hermite
0:03:58.239,0:04:01.519
polynomials again almost certainly
0:03:59.760,0:04:04.480
pronounced differently to that
0:04:01.519,0:04:05.519
so they look like this so they're
0:04:04.480,0:04:08.480
defined by
0:04:05.519,0:04:09.280
this expression involving powers of the
0:04:08.480,0:04:11.439
derivative
0:04:09.280,0:04:12.959
the total derivative with respect to
0:04:11.439,0:04:16.000
y
0:04:12.959,0:04:17.440
the first few look like this
0:04:16.000,0:04:19.440
and so on and you can work them out
0:04:17.440,0:04:22.639
yourself and these
0:04:19.440,0:04:25.120
hermite polynomials are eigen functions
0:04:22.639,0:04:26.400
of that differential equation, the hermite
0:04:25.120,0:04:27.280
equation, that we saw on the previous
0:04:26.400,0:04:29.280
board
0:04:27.280,0:04:30.880
and the corresponding eigenvalues are as
0:04:29.280,0:04:33.840
follows
0:04:30.880,0:04:34.479
that is they're just the odd numbers so
0:04:34.479,0:04:38.240
there are a couple of important
0:04:35.680,0:04:40.560
properties of these polynomials which
0:04:38.240,0:04:42.400
we'll look at now
0:04:40.560,0:04:45.040
so the first is that they are orthogonal
0:04:42.400,0:04:45.040
to one another
0:04:45.199,0:04:49.440
that is the inner product to find the
0:04:48.320,0:04:52.960
functions
0:04:49.440,0:04:54.720
of H_m and H_n is
0:04:52.960,0:04:56.240
some prefactor multiplying the Kronecker
0:04:54.720,0:04:58.960
delta
0:04:56.240,0:04:59.919
so if n doesn't equal m the inner
0:04:58.960,0:05:03.199
product is zero
0:04:59.919,0:05:06.320
if n equals m it's one multiplied by
0:05:03.199,0:05:09.120
a normalization factor
0:05:06.320,0:05:10.560
the only major difference here is this
0:05:09.120,0:05:13.120
object here and this is what's called
0:05:10.560,0:05:14.560
the weight function
0:05:13.120,0:05:16.320
so in general the inner product between
0:05:14.560,0:05:18.960
two functions
0:05:16.320,0:05:19.840
can have a weight function like this
0:05:18.960,0:05:21.600
this happens
0:05:19.840,0:05:23.039
to be useful in many cases and in this
0:05:21.600,0:05:24.080
case the weight function happens to be a
0:05:23.039,0:05:26.400
Gaussian
0:05:24.080,0:05:28.639
so this is all worked out by hermite and
0:05:26.400,0:05:30.960
others a long time ago
0:05:28.639,0:05:32.960
another important property of these
0:05:30.960,0:05:35.919
hermite functions is that they form a
0:05:32.960,0:05:35.919
complete basis.
0:05:36.000,0:05:40.560
They form a
0:05:38.320,0:05:42.560
complete orthogonal basis
0:05:40.560,0:05:43.600
for functions f of x satisfying this
0:05:42.560,0:05:45.120
property
0:05:43.600,0:05:46.880
but actually we knew this from our
0:05:45.120,0:05:49.360
postulates of quantum mechanics
0:05:46.880,0:05:50.320
it's an orthogonal basis rather than
0:05:49.360,0:05:52.880
orthonormal
0:05:50.320,0:05:53.759
just because the normalization is a
0:05:52.880,0:05:55.919
bit off here
0:05:53.759,0:05:57.600
we'll substitute back into solve the
0:05:55.919,0:05:58.560
full quantum problem in a second
0:05:57.600,0:06:00.800
and then we'll have a complete
0:05:58.560,0:06:02.160
orthonormal basis but this is one of
0:06:00.800,0:06:03.600
our postulates so the fact that it's
0:06:02.160,0:06:06.479
solving a quantum problem
0:06:03.600,0:06:08.479
tells us that this should be true and
0:06:06.479,0:06:11.120
this condition down here just tells us
0:06:08.479,0:06:13.280
that the functions we're considering
0:06:11.120,0:06:15.039
must be normalizable in this sense
0:06:13.280,0:06:16.319
but again we need that because our
0:06:15.039,0:06:18.560
functions
0:06:16.319,0:06:19.840
should be living in hilbert space and
0:06:18.560,0:06:20.479
part of that definition is that they're
0:06:20.479,0:06:25.520
square integrable so that they give
0:06:23.520,0:06:27.199
normalizable wave functions
0:06:25.520,0:06:29.039
okay so let's substitute it back in to
0:06:27.199,0:06:32.160
look at the solution to the
0:06:29.039,0:06:34.560
harmonic oscillator
0:06:32.160,0:06:36.400
so the energies as promised are
0:06:34.560,0:06:39.199
evenly spaced in energy
0:06:36.400,0:06:40.560
you'll notice that the ground state
0:06:39.199,0:06:42.560
has n equals zero
0:06:40.560,0:06:43.840
but it does not have zero energy so
0:06:42.560,0:06:45.600
there's what's called a ground state
0:06:43.840,0:06:47.600
energy of this system
0:06:45.600,0:06:49.599
the minimum energy a particle can have
0:06:47.600,0:06:52.800
in this quadratic potential
0:06:49.599,0:06:54.400
is not zero it's h bar omega over two
0:06:52.800,0:06:56.319
this is also sometimes called a zero
0:06:54.400,0:06:58.080
point energy for the system
0:06:56.319,0:07:00.560
it's somewhat mysterious but not too
0:06:58.080,0:07:02.319
much of a problem if you think about it
0:07:00.560,0:07:03.919
the potential energy of something you
0:07:02.319,0:07:05.520
can increase all the potential energies
0:07:03.919,0:07:07.599
in the universe by the same amount and
0:07:05.520,0:07:09.360
that wouldn't be an observable effect
0:07:07.599,0:07:10.639
so we have methods of dealing with this
0:07:09.360,0:07:14.479
when it comes up
0:07:10.639,0:07:16.800
in other problems the
0:07:14.479,0:07:18.560
the solutions the eigenfunctions take
0:07:16.800,0:07:21.039
the form of hermite polynomials
0:07:18.560,0:07:22.639
being multiplied by some of the bits
0:07:21.039,0:07:24.400
and pieces
0:07:22.639,0:07:26.000
these now form a complete orthonormal
0:07:24.400,0:07:27.599
basis
0:07:26.000,0:07:29.440
and more generally if we look at the
0:07:27.599,0:07:31.120
equation itself
0:07:29.440,0:07:32.880
for the the time independent Schrodinger
0:07:31.120,0:07:34.560
equation here it reminds us that
0:07:32.880,0:07:36.319
while we've just put this potential in a
0:07:34.560,0:07:38.160
quadratic potential what we're really
0:07:36.319,0:07:40.319
looking at here is the quantum version
0:07:38.160,0:07:42.080
of a spring oscillating
0:07:40.319,0:07:43.360
which would be described by a
0:07:42.080,0:07:45.199
potential like this
0:07:43.360,0:07:46.879
or any of the set of problems which we
0:07:45.199,0:07:49.360
approximate as being like that
0:07:46.879,0:07:50.960
so for example a pendulum swinging at
0:07:49.360,0:07:52.960
small angles
0:07:50.960,0:07:55.039
but again we use it quite
0:07:52.960,0:07:56.240
ubiquitously the the reason the spring
0:07:55.039,0:07:57.360
is such an important problem in
0:07:56.240,0:08:00.560
classical mechanics
0:07:57.360,0:08:03.599
is again that we can approximate many
0:08:00.560,0:08:04.879
functions by the quadratic points the
0:08:03.599,0:08:08.400
relevant bits
0:08:04.879,0:08:10.080
of interest at the maximum minima okay
0:08:08.400,0:08:11.360
so in the next video we'll take a look at
0:08:10.080,0:08:12.960
solving this again using a different
0:08:11.360,0:08:16.560
method based on operators
0:08:12.960,0:08:16.560
thanks for your time
V8.2a Ladder operators (Part 1)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
introducing ladder (raising/lowering, creation/annihilation) operators for the quantum harmonic oscillator. Writing the Hamiltonian in terms of ladder operators; commutation relations for the ladder operators.
0:00:00.160,0:00:04.560
hello in this video we're going to take
0:00:02.080,0:00:06.399
another look at the harmonic oscillator
0:00:04.560,0:00:08.240
which we looked at in the previous
0:00:06.399,0:00:09.120
video but this time we're going to look
0:00:08.240,0:00:11.759
at it using
0:00:09.120,0:00:12.559
operator methods which is a lot more
0:00:11.759,0:00:15.360
exciting
0:00:12.559,0:00:16.160
okay so here is the the time
0:00:15.360,0:00:20.400
independent Schrodinger
0:00:16.160,0:00:21.439
equation so let's write the kinetic
0:00:20.400,0:00:23.680
energy term as p
0:00:21.439,0:00:24.640
squared over 2m with p the momentum
0:00:23.680,0:00:26.720
operator
0:00:24.640,0:00:28.720
plus half m omega squared x squared
0:00:26.720,0:00:31.599
where x is the position operator
0:00:28.720,0:00:33.520
and it's acting on eigenstates phi_n and
0:00:31.599,0:00:35.440
giving eigen energies E_n
0:00:33.520,0:00:37.600
and again the phi isn't really serving a
0:00:35.440,0:00:40.239
purpose here let's just label this ket
0:00:37.600,0:00:40.239
as |n>
0:00:41.120,0:00:44.320
and bear in mind that the wave function
0:00:43.520,0:00:47.520
we solved for
0:00:44.320,0:00:50.079
in the previous video which we call phi_n(x)
0:00:51.840,0:00:56.399
here that's just given by the x
0:00:53.840,0:00:56.399
projection
0:00:56.879,0:01:00.320
of these states we're about to solve but
0:00:58.960,0:01:03.760
let's keep working with
0:01:00.320,0:01:05.519
the ket formalism okay so
0:01:03.760,0:01:06.880
a very convenient trick that we can
0:01:05.519,0:01:08.560
use in this problem and actually many
0:01:06.880,0:01:10.479
problems is to define the following
0:01:08.560,0:01:12.960
operators
0:01:10.479,0:01:15.040
so we define an operator a^dagger to be
0:01:12.960,0:01:17.040
this it's a linear combination of the x
0:01:15.040,0:01:20.560
operator and p operator
0:01:17.040,0:01:21.439
and its Hermitian conjugate given by
0:01:20.560,0:01:24.400
this
0:01:21.439,0:01:26.400
remember that p and x are both
0:01:24.400,0:01:27.759
Hermitian themselves so
0:01:26.400,0:01:30.640
they return to themselves under Hermitian
0:01:27.759,0:01:31.840
conjugation so these two operators have
0:01:30.640,0:01:36.000
various names
0:01:31.840,0:01:38.720
they're called ladder operators
0:01:36.000,0:01:39.920
raising and lowering operators where
0:01:38.720,0:01:40.880
this is the raising and this is the
0:01:39.920,0:01:42.880
lowering
0:01:40.880,0:01:44.399
and creation or annihilation
0:01:42.880,0:01:46.159
operators
0:01:44.399,0:01:47.600
where the last term will hopefully
0:01:46.159,0:01:50.159
make more sense by the
0:01:47.600,0:01:51.600
end of this set of videos so we can
0:01:50.159,0:01:53.520
see straight away that these do not
0:01:51.600,0:01:54.560
correspond to physical observables
0:01:53.520,0:01:56.560
because these operators are not
0:01:54.560,0:01:58.799
hermitian a and a^dagger
0:01:56.560,0:02:00.079
are different operators but they are
0:01:58.799,0:02:02.719
nevertheless useful
0:02:00.079,0:02:04.719
so let's put them to that use now as
0:02:02.719,0:02:07.119
follows
0:02:04.719,0:02:08.080
so first consider what we get when we
0:02:07.119,0:02:11.200
have a^dagger
0:02:08.080,0:02:12.480
it must be given by this we can
0:02:11.200,0:02:14.239
multiply this out
0:02:12.480,0:02:15.840
so we're going to get four terms we'll
0:02:14.239,0:02:19.360
get this term
0:02:15.840,0:02:21.760
this term this term and this term just
0:02:19.360,0:02:24.959
like multiplying out any quadratic
0:02:21.760,0:02:24.959
and this gives us the following
0:02:25.360,0:02:30.080
where we've been careful to keep track
0:02:27.360,0:02:31.599
of the non-commuting operators x and p
0:02:30.080,0:02:35.440
and in fact this expression here is
0:02:31.599,0:02:35.440
simply the commutator of those two terms
0:02:35.519,0:02:39.760
but we know that by definition our
0:02:37.840,0:02:42.800
canonical commutation relation
0:02:39.760,0:02:45.680
is that the commutator of x and p is
0:02:42.800,0:02:48.160
i h bar multiplying the identity
0:02:45.680,0:02:50.080
operator
0:02:48.160,0:02:52.480
and so multiplying through we find the
0:02:50.080,0:02:52.480
result
0:02:52.560,0:02:59.519
multiplying both sides by
0:02:55.840,0:02:59.519
h bar omega we find this
0:03:00.080,0:03:04.159
so this bit is just our time
0:03:02.640,0:03:05.440
independent schrodinger equation this is
0:03:04.159,0:03:08.000
our hamiltonian
0:03:05.440,0:03:09.599
we've got this extra term added here so
0:03:08.000,0:03:11.440
rearranging a bit we find the following
0:03:09.599,0:03:13.120
result
0:03:11.440,0:03:15.760
for our hamiltonian and for our
0:03:13.120,0:03:17.200
time-independent schrodinger equation
0:03:15.760,0:03:19.360
well we can actually drop the identity
0:03:17.200,0:03:22.800
operator now because this
0:03:19.360,0:03:22.800
is just the number one
0:03:22.879,0:03:27.040
and so this is
0:03:25.599,0:03:28.560
then our time independent Schrodinger
0:03:27.040,0:03:30.480
equation
0:03:28.560,0:03:32.159
so let's consider this commutator it's
0:03:30.480,0:03:34.159
given by this
0:03:32.159,0:03:35.519
just inserting the forms of a and
0:03:34.159,0:03:38.239
a^dagger of course
0:03:35.519,0:03:39.519
x is going to commute with x p with p so
0:03:38.239,0:03:42.080
we only get the
0:03:39.519,0:03:45.120
x with p commutator terms and so it
0:03:42.080,0:03:45.120
reduces to the following
0:03:45.280,0:03:49.440
where this cancels with this this
0:03:48.000,0:03:52.879
cancels with this
0:03:49.440,0:03:55.599
and p commutator x is minus
0:03:52.879,0:03:56.159
i h bar times the identity operator and
0:03:55.599,0:03:59.439
so the
0:03:56.159,0:04:01.920
end result is this the commutator of
0:03:59.439,0:04:02.959
a^dagger is the identity operator all
0:04:01.920,0:04:05.040
right that's good
0:04:02.959,0:04:06.640
and the really important result is
0:04:05.040,0:04:09.519
when we now check the commutator with
0:04:06.640,0:04:09.519
the hamiltonian
0:04:10.080,0:04:13.200
so the commutator of the hamiltonian
0:04:11.840,0:04:14.480
with a^dagger
0:04:13.200,0:04:16.479
is given by this because remember the
0:04:14.480,0:04:18.560
hamiltonian is just h bar omega
0:04:16.479,0:04:20.079
multiplying a^dagger a plus a half times
0:04:18.560,0:04:21.359
the identity operator
0:04:20.079,0:04:23.840
by definition this commutes with
0:04:21.359,0:04:25.199
everything so this disappears
0:04:23.840,0:04:26.560
so we're left with this commutator to
0:04:25.199,0:04:27.680
evaluate but this is quite
0:04:26.560,0:04:28.240
straightforward because we can just
0:04:27.680,0:04:31.680
multiply
0:04:28.240,0:04:33.360
we can just expand it that's just
0:04:31.680,0:04:34.960
writing the commutator out
0:04:33.360,0:04:36.400
and we notice that we have an
0:04:34.960,0:04:38.960
a^dagger on the left of everything so we
0:04:36.400,0:04:40.880
can write it as follows
0:04:38.960,0:04:42.960
this is just the commutator of a with
0:04:40.880,0:04:44.560
a^dagger
0:04:42.960,0:04:47.040
which we know is just the identity
0:04:44.560,0:04:47.040
operator
0:04:47.520,0:04:54.080
and so this is our result similarly for
0:04:51.040,0:04:54.080
h with a
0:04:54.240,0:05:00.880
we find just minus h bar omega times a
0:04:58.880,0:05:02.560
okay so that's all well and good but now
0:05:00.880,0:05:04.160
we can use it to actually derive the
0:05:02.560,0:05:06.720
full set of solutions for the energy
0:05:04.160,0:05:10.479
eigenstates and the energy eigenvalues
0:05:06.720,0:05:14.080
so let's do that now
0:05:10.479,0:05:14.080
actually let's do it to the next video
0:05:16.240,0:05:25.840
what are you up to
V8.2b Ladder operators (Part 2)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
using the ladder operators to derive the energy eigenstates and eigenvalues of the quantum harmonic oscillator.
0:00:00.160,0:00:03.679
hello in this video we're going to
0:00:01.920,0:00:06.080
derive the energy eigenstates and
0:00:03.679,0:00:09.760
eigenvalues of the harmonic oscillator
0:00:06.080,0:00:12.160
using ladder operators now what happens
0:00:09.760,0:00:14.080
if we act from the left with the state
0:00:12.160,0:00:16.960
a^dagger
0:00:14.080,0:00:18.000
we get this but we can move the
0:00:16.960,0:00:19.439
a^dagger through the H
0:00:18.000,0:00:21.199
using the commutator because we can
0:00:19.439,0:00:24.720
rewrite a^dagger H
0:00:21.199,0:00:26.640
as follows H a^dagger
0:00:24.720,0:00:28.320
plus the commutator of a^dagger H
0:00:26.640,0:00:32.079
because if you multiply this out you get
0:00:28.320,0:00:33.200
the a^dagger H back minus the H a^dagger
0:00:32.079,0:00:34.480
which comes here until we get the
0:00:33.200,0:00:36.719
original form
0:00:34.480,0:00:39.360
but a^dagger commutator with H we've
0:00:36.719,0:00:42.480
just worked out
0:00:39.360,0:00:44.079
and it's this we worked out H a
0:00:42.480,0:00:45.440
dagger on the previous board and found h
0:00:44.079,0:00:49.039
bar omega a^dagger
0:00:45.440,0:00:50.000
so a^dagger commutator H is minus
0:00:49.039,0:00:53.600
that
0:00:50.000,0:00:54.320
so we can add this term over to the
0:00:53.600,0:00:56.000
other side
0:00:54.320,0:00:58.160
because it's just an a^dagger
0:00:56.000,0:01:01.680
multiplying a ket n
0:00:58.160,0:01:01.680
and this gives us the result
0:01:02.239,0:01:06.640
okay so this is our result
0:01:04.640,0:01:08.159
first
0:01:06.640,0:01:10.320
let's take a look at it what it's
0:01:08.159,0:01:12.560
telling us is that we know
0:01:10.320,0:01:14.479
so we define n to be an eigen state of
0:01:12.560,0:01:16.720
the hamiltonian with energy E_n
0:01:14.479,0:01:18.159
and this tells us that the object in
0:01:16.720,0:01:19.119
parentheses because it's the same on
0:01:18.159,0:01:20.960
both sides
0:01:19.119,0:01:23.360
must also be an eigen state of the
0:01:20.960,0:01:26.720
hamiltonian but this one has the energy
0:01:23.360,0:01:28.720
E_n plus H bar omega and so what this is
0:01:26.720,0:01:29.840
telling us is that acting a^dagger on
0:01:28.720,0:01:31.280
the state n
0:01:29.840,0:01:33.200
gives us something proportional to the
0:01:31.280,0:01:35.280
state n plus one
0:01:33.200,0:01:37.119
okay it's just increased the energy of
0:01:35.280,0:01:38.240
the system and you can keep doing this
0:01:37.119,0:01:40.000
repeatedly
0:01:38.240,0:01:42.159
and you'll find that acting a^dagger
0:01:40.000,0:01:43.200
multiple times raises your energy and
0:01:42.159,0:01:46.159
each time it raises it
0:01:43.200,0:01:48.240
by a fixed amount H bar omega so just as
0:01:46.159,0:01:50.799
we saw in the differential equation case
0:01:48.240,0:01:52.320
there's an infinite ladder of
0:01:50.799,0:01:56.000
states evenly spaced in
0:01:52.320,0:01:59.200
energy so we can state the following
0:01:56.000,0:02:00.000
a^dagger acted m times on |n> on the
0:01:59.200,0:02:01.759
state |n>
0:02:00.000,0:02:03.280
is a state proportional to |n+m>
0:02:01.759,0:02:05.439
we'll find this
0:02:03.280,0:02:06.640
normalization shortly and similarly
0:02:05.439,0:02:08.160
actually if you make the same reasoning
0:02:06.640,0:02:09.679
with a instead of a^dagger
0:02:08.160,0:02:11.680
you find that that acts as a lowering
0:02:09.679,0:02:12.959
operator it lowers the energy
0:02:11.680,0:02:14.080
each time it gives you back another
0:02:14.080,0:02:16.800
eigenstate
0:02:15.040,0:02:18.560
with an energy which is lowered by H bar
0:02:16.800,0:02:20.480
omega
0:02:18.560,0:02:21.760
so we're getting there with our solution
0:02:20.480,0:02:22.640
we know that there's an infinite ladder
0:02:21.760,0:02:24.800
of states
0:02:22.640,0:02:25.920
separated by energies hbar omega we
0:02:24.800,0:02:29.280
need to work out what
0:02:25.920,0:02:29.280
those energies actually are
0:02:29.440,0:02:33.519
so first of all we need to look for the
0:02:31.599,0:02:35.040
ground state of the system that is the
0:02:33.519,0:02:36.160
state with the lowest energy
0:02:35.040,0:02:37.680
so even though there's an infinite
0:02:36.160,0:02:38.800
number of rungs in our ladder there is a
0:02:37.680,0:02:40.319
bottom rung
0:02:38.800,0:02:41.840
and let's find that now let's prove
0:02:40.319,0:02:44.959
that's true and then find it
0:02:41.840,0:02:47.840
so first the proof so first of all
0:02:44.959,0:02:50.319
we know that for any vector whatsoever
0:02:47.840,0:02:52.959
and by extension any state
0:02:50.319,0:02:53.360
so any state in here the norm square of
0:02:52.959,0:02:55.599
that
0:02:53.360,0:02:57.040
must be greater than or equal to zero
0:02:55.599,0:02:59.120
and it's equal to zero
0:02:57.040,0:03:00.640
if and only if the state itself is just
0:02:59.120,0:03:03.840
the number zero
0:03:00.640,0:03:04.159
so that's true for the state a acting
0:03:03.840,0:03:08.959
on
0:03:04.159,0:03:11.920
|n> but we can rewrite this as follows
0:03:08.959,0:03:12.480
because the norm squared of a ket any
0:03:11.920,0:03:15.440
ket
0:03:12.480,0:03:17.519
is just the bra acting on the ket the
0:03:15.440,0:03:21.200
bracket
0:03:17.519,0:03:22.560
so it's this but we know that this is
0:03:21.200,0:03:24.959
related to our hamiltonian by the
0:03:22.560,0:03:27.440
following expression
0:03:24.959,0:03:29.200
so it's hamiltonian over hbar
0:03:27.440,0:03:30.959
omega minus a half times the identity
0:03:29.200,0:03:32.640
operator
0:03:30.959,0:03:35.920
but the hamiltonian acting on an eigen
0:03:32.640,0:03:35.920
state n is just E_n
0:03:38.799,0:03:43.680
and so what we see for and the identity
0:03:41.519,0:03:46.640
operator acting on any state including n
0:03:43.680,0:03:48.400
is just one and so we see that this
0:03:46.640,0:03:50.080
quantity
0:03:48.400,0:03:52.239
must be greater than or equal to zero
0:03:50.080,0:03:55.120
that is
0:03:52.239,0:03:57.040
E_n is greater than or equal to H bar
0:03:55.120,0:03:58.480
omega over two
0:03:57.040,0:04:00.319
so we haven't proven that this is the
0:03:58.480,0:04:01.920
ground state energy yet but what we have
0:04:00.319,0:04:02.560
proven is that there's a bottom rung of
0:04:01.920,0:04:04.080
the ladder
0:04:02.560,0:04:05.599
because there's a lowest possible energy
0:04:04.080,0:04:07.599
the energies can't go below this and
0:04:05.599,0:04:10.560
that means there must be a bottom rung
0:04:07.599,0:04:11.280
so then there must exist a state a
0:04:10.560,0:04:14.159
acting on
0:04:11.280,0:04:14.959
|n> which gives us literally the value
0:04:14.159,0:04:18.799
zero
0:04:14.959,0:04:21.359
okay so the existence of
0:04:18.799,0:04:23.199
a bottom of the rung of the ladder means
0:04:21.359,0:04:25.280
that if we act the annihilation operator
0:04:23.199,0:04:28.400
the lowering operator
0:04:25.280,0:04:29.840
on that state sorry this is
0:04:28.400,0:04:32.880
a particular n now
0:04:29.840,0:04:34.320
we'll call it state n equals zero
0:04:32.880,0:04:35.919
so this is a ket zero it's not the
0:04:34.320,0:04:38.080
number zero it's just
0:04:35.919,0:04:39.440
a ket it's a state in the hilbert space
0:04:38.080,0:04:41.280
labeled by
0:04:39.440,0:04:42.960
the number zero because it's going to
0:04:41.280,0:04:45.120
be the ground state of the system
0:04:42.960,0:04:46.880
and by definition in fact this defines
0:04:45.120,0:04:48.639
this state zero
0:04:46.880,0:04:50.639
acting the lowering operator on that
0:04:48.639,0:04:51.840
state gives literally the number zero so
0:04:50.639,0:04:52.639
this isn't just a label this is the
0:04:51.840,0:04:55.199
number zero
0:04:52.639,0:04:56.320
whereas this one is n equals zero we
0:04:55.199,0:04:59.440
could also call this
0:04:56.320,0:04:59.440
if it's less confusing
0:05:00.080,0:05:03.120
we could go back to calling it phi then
0:05:02.080,0:05:04.639
it would be phi zero
0:05:03.120,0:05:06.160
if that's a bit easier but it's just the
0:05:04.639,0:05:10.400
state zero
0:05:06.160,0:05:12.000
okay so
0:05:10.400,0:05:13.280
there must be a state which has this
0:05:12.000,0:05:14.320
property and that we can use this to
0:05:13.280,0:05:17.520
find the ground state
0:05:14.320,0:05:19.199
wave function of the system
0:05:17.520,0:05:21.120
so starting from the defining equation
0:05:19.199,0:05:24.639
of our ground state
0:05:21.120,0:05:27.280
label zero we can rewrite
0:05:24.639,0:05:28.560
things into the
0:05:27.280,0:05:32.400
position basis
0:05:28.560,0:05:34.240
by expanding our operator a
0:05:32.400,0:05:35.840
so first written out in terms of
0:05:34.240,0:05:36.560
position and momentum operators we have
0:05:35.840,0:05:38.639
this
0:05:36.560,0:05:41.120
let's write things into the position
0:05:38.639,0:05:41.120
basis
0:05:41.520,0:05:44.639
so the moment the position operator
0:05:44.240,0:05:48.479
is just
0:05:44.639,0:05:51.680
x the momentum operator is p is minus i
0:05:48.479,0:05:52.080
h bar d by dx which i've substituted in
0:05:51.680,0:05:53.759
here
0:05:52.080,0:05:55.440
d by dx can be a total derivative
0:05:53.759,0:05:56.160
because x is the only variable in the
0:05:55.440,0:05:57.919
problem
0:05:56.160,0:05:59.199
and the ground state zero projected
0:05:57.919,0:06:01.680
into the x basis
0:05:59.199,0:06:02.400
let's write it phi zero of x we can't
0:06:01.680,0:06:04.400
really write it
0:06:02.400,0:06:06.319
just zero of x doesn't really work so
0:06:04.400,0:06:08.080
another strength of the ket
0:06:06.319,0:06:09.600
formalism is that we can give arbitrary
0:06:08.080,0:06:10.800
labels like that but when it comes to
0:06:09.600,0:06:12.479
wave functions we want to use something
0:06:10.800,0:06:13.680
like phi
0:06:12.479,0:06:15.199
okay well this is a simple
0:06:13.680,0:06:15.600
differential equation to solve and the
0:06:15.199,0:06:19.520
result
0:06:15.600,0:06:22.560
it looks like this so a gaussian
0:06:19.520,0:06:25.120
this is the ground state wave function
0:06:22.560,0:06:26.319
and to find the ground state itself the
0:06:25.120,0:06:28.479
energy eigenvalue
0:06:26.319,0:06:30.479
we just substitute this back in to our
0:06:28.479,0:06:33.680
time independent Schrodinger equation
0:06:30.479,0:06:34.639
and we find the result so this is our
0:06:33.680,0:06:36.319
time-independent
0:06:34.639,0:06:38.800
schrodinger equation this is our
0:06:36.319,0:06:40.560
hamiltonian
0:06:38.800,0:06:43.120
so we write it out and we find that
0:06:40.560,0:06:43.840
we have the energy eigenvalue associated
0:06:43.120,0:06:45.919
with this
0:06:43.840,0:06:47.520
ground state wave function is hbar
0:06:45.919,0:06:49.120
omega over two
0:06:47.520,0:06:51.280
so actually the bound that we saw on the
0:06:49.120,0:06:53.039
previous board is saturated
0:06:51.280,0:06:54.960
and this is the ground state energy or
0:06:53.039,0:06:56.479
the zero point energy we found the
0:06:54.960,0:06:58.240
bottom of the rum bottom rung of the
0:06:56.479,0:07:00.240
ladder
0:06:58.240,0:07:02.639
so the ground state energy or the zero
0:07:00.240,0:07:05.120
point energy we denote it E_0
0:07:02.639,0:07:07.199
and we can find now all the other energy
0:07:05.120,0:07:09.360
eigen states and eigen values
0:07:07.199,0:07:11.440
using our raising operators starting
0:07:09.360,0:07:14.639
from this lowest rung
0:07:11.440,0:07:17.840
okay so let's leave it at that for
0:07:14.639,0:07:17.840
this video thank you for your time
V8.3 The number operator
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
rewriting the quantum harmonic oscillator Hamiltonian in terms of the number operator; deriving the normalisation of the states acted upon by raising an lowering operators.
0:00:00.399,0:00:03.520
hello in this video we're going to
0:00:01.920,0:00:05.440
continue looking at properties of the
0:00:03.520,0:00:06.720
ladder operators associated with the
0:00:05.440,0:00:07.839
Harmonic oscillator
0:00:06.720,0:00:10.639
in particular we'll look at something
0:00:07.839,0:00:12.639
called the number operator
0:00:10.639,0:00:14.559
so in the previous video we found the
0:00:12.639,0:00:16.880
ground state of the system
0:00:14.559,0:00:18.560
we found that the time
0:00:16.880,0:00:21.680
independent Schrodinger equation
0:00:18.560,0:00:23.680
for the ground state which we define
0:00:21.680,0:00:25.279
to be this ket with a zero in it zero is
0:00:23.680,0:00:26.960
just a label telling instead of the
0:00:25.279,0:00:28.480
ground state this is not the number zero
0:00:26.960,0:00:31.279
it's the state zero
0:00:28.480,0:00:33.360
gives us the value E_0 which is hbar
0:00:31.279,0:00:36.480
omega over two
0:00:33.360,0:00:36.480
and we also found that
0:00:36.559,0:00:40.000
we found that for any state |n> which
0:00:39.200,0:00:43.200
solves the time
0:00:40.000,0:00:45.840
independent schrodinger equation with
0:00:43.200,0:00:46.960
energy eigenvalue E_n if we act the
0:00:45.840,0:00:49.039
raising operator
0:00:46.960,0:00:50.480
a^dagger m times where m is just a
0:00:49.039,0:00:51.760
positive integer it's nothing to do with
0:00:50.480,0:00:54.320
the mass
0:00:51.760,0:00:55.120
then we get another eigenstate
0:00:54.320,0:00:57.199
of the system
0:00:55.120,0:00:58.640
defined by this equation which has
0:00:57.199,0:01:01.120
energy E_n
0:00:58.640,0:01:02.160
plus m h bar omega again m being the
0:01:01.120,0:01:05.040
integer
0:01:02.160,0:01:06.400
so what this tells us is that we can
0:01:05.040,0:01:08.640
write down state
0:01:06.400,0:01:10.080
n plus m in here again this is always
0:01:08.640,0:01:10.960
just a label inside the ket but this is
0:01:10.080,0:01:12.720
a convenient one
0:01:10.960,0:01:15.119
because it's an energy eigenstate with
0:01:12.720,0:01:17.759
energy eigenvalue n plus m where these
0:01:15.119,0:01:19.200
are the two relevant integers okay
0:01:17.759,0:01:20.560
putting this together we see that we
0:01:19.200,0:01:22.560
have the ground state we have an
0:01:20.560,0:01:24.159
infinite ladder of excited states
0:01:22.560,0:01:25.840
evenly spaced in energy above that
0:01:24.159,0:01:28.080
ground state and so we get the total
0:01:25.840,0:01:31.439
solution to the problem
0:01:28.080,0:01:33.520
the hamiltonian acting on eigenstate n
0:01:31.439,0:01:35.200
is just equal to h bar omega times
0:01:33.520,0:01:37.439
a^dagger a plus a half
0:01:35.200,0:01:39.040
acting on n we've seen this before in
0:01:37.439,0:01:40.880
the previous video
0:01:39.040,0:01:43.040
and what this is telling us is that the
0:01:40.880,0:01:45.920
energy eigen value associated with that
0:01:43.040,0:01:46.320
is just h bar omega n plus a half where
0:01:45.920,0:01:48.560
n
0:01:46.320,0:01:49.520
is an integer labeling the state and
0:01:48.560,0:01:52.960
then here goes
0:01:49.520,0:01:54.960
from
0:01:52.960,0:01:57.520
zero up so zero is the energy of
0:01:54.960,0:01:59.360
the ground state
0:01:57.520,0:02:01.040
so what this is telling us if we look
0:01:59.360,0:02:02.560
at it so this is this is a
0:02:01.040,0:02:03.840
hamiltonian this is a hamiltonian
0:02:02.560,0:02:06.560
rewritten and this is an energy
0:02:03.840,0:02:06.560
eigenvalue
0:02:06.719,0:02:09.840
so actually there's a simpler operator
0:02:08.399,0:02:12.000
that we can define here which is as
0:02:09.840,0:02:13.200
follows
0:02:12.000,0:02:15.120
you see that the only difference in
0:02:13.200,0:02:16.800
these two equations is that this
0:02:15.120,0:02:19.760
operator a^dagger a
0:02:16.800,0:02:21.200
has returned this integer n when acted
0:02:19.760,0:02:23.040
on the state |n>
0:02:21.200,0:02:24.800
which is the state corresponding to the
0:02:23.040,0:02:26.879
nth energy eigen state
0:02:24.800,0:02:28.160
and so we define a^dagger a to be this
0:02:26.879,0:02:30.160
operator n
0:02:28.160,0:02:31.920
the number operator whose defining
0:02:30.160,0:02:34.400
equation is this
0:02:31.920,0:02:35.519
the operator n acting on eigenstate n
0:02:34.400,0:02:36.560
and it's an eigenstate of the
0:02:35.519,0:02:38.959
hamiltonian
0:02:36.560,0:02:40.560
gives us back the integer n labeling
0:02:38.959,0:02:42.480
the the number of the state
0:02:40.560,0:02:45.680
times the state so it's an eigenvalue
0:02:42.480,0:02:45.680
equation for n
0:02:45.840,0:02:51.760
so this turns out to be a useful
0:02:47.120,0:02:53.599
operator to work with in many scenarios
0:02:51.760,0:02:55.360
let's take a look at the normalization
0:02:53.599,0:02:57.920
associated with these operators
0:02:55.360,0:02:58.959
so we know the following relations we
0:02:57.920,0:03:01.599
know that a^dagger
0:02:58.959,0:03:02.959
raises the eigen state to the next eigen
0:03:01.599,0:03:05.360
state up in the ladder
0:03:02.959,0:03:08.080
and the operator a lowers the eigen
0:03:05.360,0:03:11.519
state let's just do that schematically
0:03:08.080,0:03:13.599
that is we have an infinite ladder of
0:03:11.519,0:03:15.280
energy eigenstates at different energies
0:03:13.599,0:03:16.879
the rungs of the ladder are evenly
0:03:15.280,0:03:18.159
spaced by hbar omega
0:03:16.879,0:03:19.519
it has a bottom rung even though
0:03:18.159,0:03:20.159
there's an infinite number of rungs
0:03:19.519,0:03:22.239
they're all
0:03:20.159,0:03:24.720
positive energy the bottom rung lies at
0:03:22.239,0:03:26.640
the energy hbar omega/2
0:03:24.720,0:03:28.159
and to starting from one rung of the
0:03:26.640,0:03:31.440
ladder one state
0:03:28.159,0:03:34.720
we can go up to the next state using
0:03:31.440,0:03:39.040
a raising operator a^dagger and we can
0:03:34.720,0:03:40.319
go back down using a lower operator a
0:03:40.319,0:03:45.120
so a^daggers take us up the ladder as
0:03:42.640,0:03:47.360
take us down the ladder
0:03:45.120,0:03:48.879
but so far we've only worked with these
0:03:47.360,0:03:51.760
proportionality signs
0:03:48.879,0:03:53.439
can we get the
0:03:51.760,0:03:55.920
constants of proportionality
0:03:53.439,0:04:01.840
out the front here and we can to do so
0:03:55.920,0:04:01.840
let's go over to the worked example area
0:04:06.560,0:04:10.799
okay so to look at the normalization of
0:04:09.680,0:04:14.159
these states
0:04:10.799,0:04:17.519
we have that a^dagger acting on n
0:04:14.159,0:04:21.040
is proportional to n plus one
0:04:17.519,0:04:23.120
and a acting on n is proportional to n
0:04:21.040,0:04:24.800
minus one
0:04:23.120,0:04:26.400
and we also have the definition of the
0:04:24.800,0:04:29.840
number operator
0:04:26.400,0:04:29.840
as a^dagger a
0:04:30.080,0:04:33.280
and we know that the number operator
0:04:31.759,0:04:35.759
acting on state n
0:04:33.280,0:04:38.000
just returns the number n because its
0:04:35.759,0:04:39.919
eigenvalue
0:04:38.000,0:04:41.120
so if we act from this on this from the
0:04:39.919,0:04:44.639
left with bra n
0:04:41.120,0:04:48.560
we find that the expectation value of n
0:04:44.639,0:04:52.240
in state n is you guessed it n
0:04:48.560,0:04:57.360
but n is a^dagger a
0:04:52.240,0:05:00.400
so this is a^dagger a
0:04:57.360,0:05:02.080
n is equal to n
0:05:00.400,0:05:04.160
but then if we look at this this is a
0:05:02.080,0:05:06.080
acting on n and this is the Hermitian
0:05:04.160,0:05:10.400
conjugate of it
0:05:06.080,0:05:13.600
and so we have that a acting on n
0:05:10.400,0:05:14.800
modulus square is equal to n
0:05:13.600,0:05:17.120
because that's just what this expression
0:05:14.800,0:05:19.680
here says
0:05:17.120,0:05:20.639
and so we have the length of the vector
0:05:19.680,0:05:24.000
a acting on
0:05:20.639,0:05:27.039
n as given by the norm is just equal
0:05:24.000,0:05:30.479
to the square root of n
0:05:27.039,0:05:32.320
okay so we have that a acting on n is
0:05:30.479,0:05:32.960
proportional to n minus one so it's n
0:05:32.320,0:05:35.759
minus one
0:05:32.960,0:05:36.800
multiplied by some pre-factor the
0:05:35.759,0:05:39.840
length of
0:05:36.800,0:05:41.919
state n minus 1
0:05:39.840,0:05:43.840
is equal to 1 because all states in the
0:05:41.919,0:05:45.520
hilbert space must be of length 1
0:05:43.840,0:05:47.199
that means they're normalized physical
0:05:45.520,0:05:49.280
states are normalized
0:05:47.199,0:05:50.479
and so if a acting on a is proportional
0:05:49.280,0:05:53.600
to n minus 1
0:05:50.479,0:05:56.400
a acting on n is of length root n
0:05:53.600,0:05:58.000
and n minus 1 itself is of the length 1
0:05:56.400,0:05:58.319
then we put it all together we see that
0:05:58.000,0:06:01.919
a
0:05:58.319,0:06:03.680
acting on n must equal the square root
0:06:01.919,0:06:07.199
of n
0:06:03.680,0:06:07.199
acting on n minus one
0:06:08.560,0:06:11.840
okay so that's the normalization of this
0:06:10.080,0:06:14.000
state let's do the same
0:06:11.840,0:06:17.360
for the raising operator and i think
0:06:14.000,0:06:17.360
i'll fit it on the same bit of paper
0:06:18.840,0:06:21.840
here
0:06:22.080,0:06:29.120
so in this case we can say that
0:06:25.120,0:06:29.120
the raising operator acting on n
0:06:29.360,0:06:38.560
modulus square
0:06:35.120,0:06:42.800
is equal to n a
0:06:38.560,0:06:44.639
a^dagger acting on n
0:06:42.800,0:06:46.880
like that because it's just the Hermitian
0:06:44.639,0:06:49.440
conjugate of this thing over here
0:06:46.880,0:06:50.800
so we need to use the commutator so this
0:06:49.440,0:06:54.160
thing equals
0:06:50.800,0:06:57.840
state n
0:06:54.160,0:07:03.840
number operator n which is a^dagger a
0:06:57.840,0:07:03.840
plus the commutator [a,a^dagger]
0:07:04.479,0:07:07.680
acting on n so if you expand this
0:07:06.479,0:07:08.880
expression here
0:07:07.680,0:07:10.960
and then you move it a bit so that you
0:07:08.880,0:07:12.960
don't get the light on so it's
0:07:10.960,0:07:14.319
if you expand this a^dagger a plus
0:07:12.960,0:07:16.479
[a,a^dagger]
0:07:14.319,0:07:19.120
you'll find that you just get a
0:07:16.479,0:07:19.120
dagger again
0:07:19.840,0:07:24.479
but this thing here is just the identity
0:07:22.560,0:07:28.080
operator
0:07:24.479,0:07:29.840
as we've seen on the board and so
0:07:28.080,0:07:31.440
acting on the state n we see that we
0:07:29.840,0:07:34.400
have
0:07:31.440,0:07:34.400
n plus one
0:07:35.680,0:07:42.000
so the length of the state
0:07:39.039,0:07:42.800
a^dagger acting on n that's given by the
0:07:42.000,0:07:47.759
norm
0:07:42.800,0:07:47.759
it's at length n plus one
0:07:47.919,0:07:52.080
square rooted and so just like before we
0:07:51.039,0:07:55.759
reasoned that
0:07:52.080,0:07:59.199
a^dagger n is equal to
0:07:55.759,0:08:01.919
n plus one square rooted
0:07:59.199,0:08:01.919
n plus one
0:08:03.360,0:08:06.319
through the same reasoning
0:08:07.520,0:08:12.560
and finally
0:08:10.560,0:08:14.879
that was all visible finally we can say
0:08:12.560,0:08:17.120
that starting from the state zero
0:08:14.879,0:08:19.280
we can act the state a we can act a
0:08:17.120,0:08:21.520
dagger on zero to get state one
0:08:19.280,0:08:22.639
and that's normalized if we apply
0:08:21.520,0:08:25.360
this chain of reasoning
0:08:22.639,0:08:26.000
from this normalization you can see
0:08:25.360,0:08:29.039
that
0:08:26.000,0:08:33.919
acting a^dagger
0:08:29.039,0:08:40.320
n times on the state 0 gives us
0:08:33.919,0:08:43.599
n factorial square rooted
0:08:40.320,0:08:43.599
times the state n
0:08:47.519,0:08:51.120
so this is just sorting the
0:08:48.800,0:08:52.720
normalization out and then the raising
0:08:51.120,0:08:54.240
and lowering operators can be used to
0:08:52.720,0:08:59.040
deduce any state
0:08:54.240,0:08:59.040
given any other okay thank you for your
0:08:59.480,0:09:02.480
time
V8.4 Second quantisation
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
recap of the quantum harmonic oscillator solutions. We can re-interpret the nth excited state as n bosons each of energy ℏω: this is called second quantisation. First quantisation is writing a wave description of particles, while second quantisation is writing a particle description of waves.
0:00:00.160,0:00:04.160
hello so we've taken a look at the
0:00:02.480,0:00:05.600
quantum harmonic oscillator
0:00:04.160,0:00:07.040
and we've studied it using ladder
0:00:05.600,0:00:08.240
operators or raising and lowering
0:00:07.040,0:00:10.000
operators
0:00:08.240,0:00:12.080
let's look a little bit at the
0:00:10.000,0:00:13.440
philosophy as to what we're doing there
0:00:12.080,0:00:15.679
and it's the philosophy of what's called
0:00:13.440,0:00:17.440
second quantization sometimes this has a
0:00:15.679,0:00:20.800
z instead of an s
0:00:17.440,0:00:22.320
so let's remind ourselves of the problem
0:00:20.800,0:00:23.920
so the time independent Schrodinger
0:00:22.320,0:00:25.920
equation reads as follows
0:00:23.920,0:00:28.240
our potential is just half m omega
0:00:25.920,0:00:29.760
squared x squared
0:00:28.240,0:00:31.439
and we can rewrite it in terms of
0:00:29.760,0:00:32.000
raising and lowering operators a^dagger
0:00:31.439,0:00:35.040
and a
0:00:32.000,0:00:36.960
in the following form where a^dagger
0:00:35.040,0:00:39.120
a here is naturally interpreted
0:00:36.960,0:00:40.000
as an operator
0:00:39.120,0:00:41.920
and
0:00:40.000,0:00:44.079
called the number operator which returns
0:00:41.920,0:00:46.960
the integer n labeling the state
0:00:44.079,0:00:48.399
where n is an integer greater than or
0:00:46.960,0:00:51.680
equal to zero
0:00:48.399,0:00:52.960
so let's draw the potential it looks
0:00:51.680,0:00:57.120
something like this
0:00:52.960,0:01:00.480
and the real parts of the
0:00:57.120,0:01:03.199
x projected eigen states at a particular
0:01:00.480,0:01:05.840
instant time look like this
0:01:03.199,0:01:06.479
so they look something like this so we
0:01:05.840,0:01:09.920
have
0:01:06.479,0:01:11.520
evenly spaced energy levels separated by
0:01:09.920,0:01:13.200
energies hbar omega
0:01:11.520,0:01:14.799
the ground state itself the lowest
0:01:13.200,0:01:18.240
energy state has only
0:01:14.799,0:01:19.920
hbar omega over two
0:01:18.240,0:01:21.680
and we can solve for the wave functions
0:01:19.920,0:01:23.280
themselves by starting from the ground
0:01:21.680,0:01:25.360
state which we can deduce
0:01:23.280,0:01:26.640
either using the Hermite polynomial
0:01:25.360,0:01:28.640
method
0:01:26.640,0:01:30.799
or we can deduce it by using the
0:01:28.640,0:01:33.280
definition of the ground state
0:01:30.799,0:01:34.560
which is that the lowering operator
0:01:33.280,0:01:37.360
acting on the ground state gives a
0:01:34.560,0:01:37.360
number zero
0:01:38.479,0:01:44.000
using that method we can deduce in
0:01:40.560,0:01:46.880
the position basis we have this
0:01:44.000,0:01:48.079
this expression a gaussian form and all
0:01:46.880,0:01:50.960
higher energy
0:01:48.079,0:01:54.720
eigenstates can be found using the
0:01:50.960,0:01:57.280
raising operator on that state
0:01:54.720,0:01:58.880
so a^dagger acting on the state n gives
0:01:57.280,0:02:00.399
square root n plus one acting on the
0:01:58.880,0:02:01.920
state n plus one and that's properly
0:02:00.399,0:02:04.240
normalized
0:02:01.920,0:02:05.680
okay but let's take a look at what
0:02:04.240,0:02:08.879
we're actually doing here
0:02:05.680,0:02:11.280
a bit more philosophically so
0:02:08.879,0:02:12.720
this operator a^dagger a we've said is
0:02:11.280,0:02:14.480
called the number operator
0:02:12.720,0:02:15.840
because it just returns the number of
0:02:14.480,0:02:19.680
the state here where this is
0:02:15.840,0:02:22.080
state 0 1 2 3 and so on
0:02:19.680,0:02:24.160
and because these and the levels of
0:02:22.080,0:02:26.160
these energies are spaced by the same
0:02:24.160,0:02:28.239
amount each time h bar omega
0:02:26.160,0:02:30.640
there's a very natural interpretation to
0:02:28.239,0:02:32.400
this not in terms of wave functions
0:02:30.640,0:02:34.800
but in terms of particles sat in this
0:02:32.400,0:02:38.400
well
0:02:34.800,0:02:40.319
so in quantum mechanics in this course
0:02:38.400,0:02:42.080
we're always dealing with single
0:02:40.319,0:02:45.680
particles at a time
0:02:42.080,0:02:46.560
but in this case it's very natural
0:02:45.680,0:02:49.680
to interpret
0:02:46.560,0:02:51.200
the nth energy level here you can
0:02:49.680,0:02:53.360
you can certainly interpret it as a wave
0:02:51.200,0:02:56.400
function which happens to have energy
0:02:53.360,0:02:57.120
h bar omega n plus half we've done that
0:02:56.400,0:02:59.840
already
0:02:57.120,0:03:02.080
but we can also interpret it as n
0:02:59.840,0:03:05.200
individual particles sat in a well
0:03:02.080,0:03:06.080
where each particle has energy h-bar
0:03:05.200,0:03:07.680
omega
0:03:06.080,0:03:10.720
plus a ground state energy that we
0:03:07.680,0:03:12.319
don't need to worry too much about
0:03:10.720,0:03:14.000
and this is the basis of what's called
0:03:12.319,0:03:16.879
second quantization
0:03:14.000,0:03:18.000
so in their quantum field theory
0:03:16.879,0:03:19.519
textbook Lancaster
0:03:18.000,0:03:21.200
and Blundell have a very nice way of
0:03:19.519,0:03:23.040
phrasing this they say the first
0:03:21.200,0:03:24.879
quantization which is what we've been
0:03:23.040,0:03:27.680
looking at at least in the first half of
0:03:24.879,0:03:29.440
this course is saying that particles
0:03:27.680,0:03:31.200
can have wave-like properties in quantum
0:03:29.440,0:03:32.480
mechanics okay we've been
0:03:31.200,0:03:34.000
describing particles using the
0:03:32.480,0:03:35.760
schrodinger equation which is a wave
0:03:34.000,0:03:37.040
equation
0:03:35.760,0:03:38.959
they say that so that's first
0:03:37.040,0:03:41.360
quantization describing particles
0:03:38.959,0:03:43.760
like waves second quantization they say
0:03:41.360,0:03:45.840
is describing waves like particles
0:03:43.760,0:03:47.280
because the same description that we're
0:03:45.840,0:03:49.200
using here
0:03:47.280,0:03:50.319
so these are wave-like things these are
0:03:49.200,0:03:52.319
wave functions
0:03:50.319,0:03:53.360
and we're saying okay here's the this
0:03:52.319,0:03:55.760
is the
0:03:53.360,0:03:56.799
so we've got zero one two three this is
0:03:55.760,0:04:00.000
the
0:03:56.799,0:04:00.720
n equals three wave function it's a form
0:04:00.000,0:04:02.400
of wave
0:04:00.720,0:04:04.000
but we could also say this is just like
0:04:02.400,0:04:08.959
three particles
0:04:04.000,0:04:11.840
each of energy h bar omega sat in a well
0:04:08.959,0:04:12.959
okay so this is saying that second
0:04:11.840,0:04:15.680
quantization
0:04:12.959,0:04:16.959
is describing waves as
0:04:15.680,0:04:18.320
particle-like
0:04:16.959,0:04:19.680
and of course that should always work we
0:04:18.320,0:04:20.799
should have this kind of wave particle
0:04:19.680,0:04:22.639
duality where we can use either
0:04:20.799,0:04:25.919
description
0:04:22.639,0:04:27.680
okay so the
0:04:25.919,0:04:29.680
properties of these particles we know
0:04:27.680,0:04:30.479
that having multiple particles in the
0:04:29.680,0:04:34.000
same
0:04:30.479,0:04:37.120
well in the same state means that
0:04:34.000,0:04:39.199
they must be bosons okay so it's n
0:04:37.120,0:04:40.720
bosonic particles in the well because we
0:04:39.199,0:04:42.639
have the
0:04:40.720,0:04:45.199
Pauli exclusion principle tells us that
0:04:42.639,0:04:46.639
no two fermions can have the same set
0:04:45.199,0:04:49.280
of quantum numbers
0:04:46.639,0:04:50.720
so it's a bosonic quantum well
0:04:49.280,0:04:52.000
there is in fact a fermionic version of
0:04:50.720,0:04:55.199
this which you can study
0:04:52.000,0:04:58.479
it's not too much harder
0:04:55.199,0:05:00.720
okay so let's recap that in bullet
0:04:58.479,0:05:02.320
points
0:05:00.720,0:05:04.479
the energy eigenvalues of the quantum
0:05:02.320,0:05:08.160
harmonic oscillator are evenly spaced
0:05:04.479,0:05:11.120
by energies h bar omega
0:05:08.160,0:05:12.720
when we have energy eigenvalue E_n we
0:05:11.120,0:05:13.840
can either think of that as the nth
0:05:12.720,0:05:17.280
excited state
0:05:13.840,0:05:20.639
within the harmonic oscillator
0:05:17.280,0:05:22.960
or we can think of it as n bosons
0:05:20.639,0:05:25.680
independent bosons of energy h bar
0:05:22.960,0:05:25.680
omega each
0:05:26.080,0:05:30.240
the ground state of the system the
0:05:27.919,0:05:31.759
lowest possible energy is not zero
0:05:30.240,0:05:34.080
it's there's a zero point energy or
0:05:31.759,0:05:38.000
ground state energy which is equal to h
0:05:34.080,0:05:41.840
bar omega over two
0:05:38.000,0:05:44.960
and we can express the eigenstate n
0:05:41.840,0:05:47.199
by the ground state zero acted on
0:05:44.960,0:05:48.560
n times by creation operators n and
0:05:47.199,0:05:53.120
similarly we can lower
0:05:48.560,0:05:56.400
the state using lowering operators
0:05:53.120,0:05:58.160
so this last point is central to why
0:05:56.400,0:06:00.240
the raising and lowering operators
0:05:58.160,0:06:02.080
are so important
0:06:00.240,0:06:03.440
they actually give us the generalization
0:06:02.080,0:06:04.319
to something we're not going to study in
0:06:03.440,0:06:06.160
this course
0:06:04.319,0:06:07.919
which is when we include a relativistic
0:06:06.160,0:06:10.880
effect into quantum mechanics
0:06:07.919,0:06:12.479
so it turns out that in order to include
0:06:10.880,0:06:13.919
special relativity and quantum mechanics
0:06:12.479,0:06:15.280
together you have to allow the number of
0:06:13.919,0:06:18.880
particles to vary
0:06:15.280,0:06:20.479
it's impossible to keep it fixed and
0:06:18.880,0:06:23.199
in order to do that we develop what's
0:06:20.479,0:06:26.960
called quantum field theory
0:06:23.199,0:06:30.240
or QFT for short so quantum field theory
0:06:26.960,0:06:32.400
treats all of space-time as
0:06:30.240,0:06:33.600
a big quantum field and depends which
0:06:32.400,0:06:34.960
kind of particle you're working with
0:06:33.600,0:06:35.759
each will have its own type of quantum
0:06:34.960,0:06:37.680
field
0:06:35.759,0:06:38.880
and the particles are excitations out of
0:06:37.680,0:06:40.560
that quantum field
0:06:38.880,0:06:42.240
and mathematically what we do is we
0:06:40.560,0:06:44.080
start with a vacuum state for the
0:06:42.240,0:06:45.520
universe so vacuum everywhere
0:06:44.080,0:06:48.080
and if we want to create a particle at
0:06:45.520,0:06:51.280
position x we use a raising operator
0:06:48.080,0:06:54.639
from the harmonic oscillator problem
0:06:51.280,0:06:55.840
and we act it at position x so we'd say
0:06:54.639,0:06:56.639
if there's a particle located at
0:06:55.840,0:06:59.360
position x
0:06:56.639,0:07:00.080
we've acted a raising operator located
0:06:59.360,0:07:02.400
at x
0:07:00.080,0:07:04.080
in quantum field theory onto the vacuum
0:07:02.400,0:07:05.120
state where this is actually often
0:07:04.080,0:07:08.160
denoted
0:07:05.120,0:07:10.319
zero so
0:07:08.160,0:07:11.759
quantum field theory describes
0:07:10.319,0:07:14.160
the universe as
0:07:11.759,0:07:15.199
made up of quantum fields for
0:07:14.160,0:07:17.120
different particles
0:07:15.199,0:07:18.479
and the particles are excitations out of
0:07:17.120,0:07:21.120
those fields
0:07:18.479,0:07:21.759
and mathematically what it does is
0:07:21.120,0:07:24.240
assign
0:07:21.759,0:07:25.919
a simple harmonic oscillator a quantum
0:07:24.240,0:07:26.800
harmonic oscillator at each point in
0:07:25.919,0:07:29.919
space
0:07:26.800,0:07:31.840
and it describes the
0:07:29.919,0:07:33.440
the emergent universe coming out of that
0:07:31.840,0:07:34.479
as excitations of those harmonic
0:07:33.440,0:07:36.720
oscillators
0:07:34.479,0:07:38.000
so by solving the harmonic oscillator
0:07:36.720,0:07:40.400
problem in quantum mechanics that's the
0:07:38.000,0:07:42.560
basis for all of quantum field theory
0:07:40.400,0:07:44.000
the simplest example of this to my mind
0:07:42.560,0:07:46.080
is when we look at sound
0:07:44.000,0:07:48.400
traveling through crystals so a crystal
0:07:46.080,0:07:50.160
has a regular periodic array of atoms
0:07:48.400,0:07:52.560
when sound travels through it it's a
0:07:50.160,0:07:54.240
vibration passing through the crystal
0:07:52.560,0:07:56.160
and of course that admits a description
0:07:54.240,0:07:58.400
in terms of waves
0:07:56.160,0:08:00.000
but second quantization allows us to
0:07:58.400,0:08:01.120
write waves in terms of particle
0:08:00.000,0:08:03.520
excitations
0:08:01.120,0:08:04.479
and so rather than saying there's a wave
0:08:03.520,0:08:06.560
with
0:08:04.479,0:08:07.840
a certain energy in this system we can
0:08:06.560,0:08:09.840
instead say that
0:08:07.840,0:08:12.080
there's a set of excitations of these
0:08:09.840,0:08:13.599
different harmonic oscillators or in
0:08:12.080,0:08:15.599
this case it's just each atom
0:08:13.599,0:08:17.680
literally vibrating about a point just
0:08:15.599,0:08:19.280
like a spring being held in place by the
0:08:17.680,0:08:21.680
neighboring atoms
0:08:19.280,0:08:22.639
and we can redescribe what in one
0:08:21.680,0:08:24.840
picture is a wave
0:08:22.639,0:08:26.479
as a set of oscillations of these
0:08:24.840,0:08:28.080
oscillators
0:08:26.479,0:08:29.599
and then we're just using this
0:08:28.080,0:08:31.840
description in terms of
0:08:29.599,0:08:33.599
bosons from the simple harmonic
0:08:31.840,0:08:37.279
oscillator picture
0:08:33.599,0:08:37.279
okay thank you for your time
V9.1 The 3D infinite potential well
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
a particle confined to an infinitely-deep potential well of cubic shape; separating into three independent 1D infinite potential well problems; eigenstates and eigenvalues; degeneracy of energy eigenstates; orthonormality of states.
0:00:00.480,0:00:04.240
hello in this video we're going to take
0:00:02.480,0:00:06.000
a look at the three-dimensional infinite
0:00:04.240,0:00:09.360
potential well
0:00:06.000,0:00:11.599
also known as the cubic box or cuboidal
0:00:09.360,0:00:15.040
box if you allow the dimensions to vary
0:00:11.599,0:00:17.039
so the potential looks like this that is
0:00:15.040,0:00:18.240
the potential is now a function of a
0:00:17.039,0:00:21.279
three-dimensional vector
0:00:18.240,0:00:22.960
x = (x,y,z) and it's zero if
0:00:21.279,0:00:25.599
all three of these conditions are met
0:00:22.960,0:00:26.640
x is between zero and l, y is
0:00:25.599,0:00:28.480
between zero and l,
0:00:26.640,0:00:31.119
and z is between zero now this is for
0:00:28.480,0:00:32.719
the cubicle box cuboidal we'd allow
0:00:31.119,0:00:35.920
these to take different values different
0:00:32.719,0:00:37.600
angles and it's infinity otherwise
0:00:35.920,0:00:40.640
so the time independent schroedinger
0:00:37.600,0:00:43.520
equation takes the following form
0:00:40.640,0:00:45.280
the hamiltonian acting on phi(x) where again
0:00:43.520,0:00:46.960
this is a three dimensional vector
0:00:45.280,0:00:49.200
it's given by minus h bar squared
0:00:46.960,0:00:51.440
over two m grad squared phi(x)
0:00:49.200,0:00:53.280
which equals the energy times phi x uh
0:00:51.440,0:00:55.520
and grad squared here can be expanded as
0:00:53.280,0:00:57.680
follows
0:00:55.520,0:00:59.520
it's the sum of three terms the
0:00:57.680,0:01:02.079
partial derivative
0:00:59.520,0:01:03.680
of phi with respect to x twice so d
0:01:02.079,0:01:06.720
squared phi by dx squared
0:01:03.680,0:01:07.520
while holding y z and t constant and the
0:01:06.720,0:01:10.960
equivalent
0:01:07.520,0:01:10.960
for y and for z
0:01:11.360,0:01:16.560
this equation is separable so
0:01:14.479,0:01:19.200
we've already separated the time part
0:01:16.560,0:01:20.880
in the usual manner but we can also
0:01:19.200,0:01:22.240
separate the x y and z parts
0:01:20.880,0:01:24.960
so let's just move this up to the top
0:01:22.240,0:01:27.680
and do that on the next board
0:01:24.960,0:01:28.560
that is we can separate the three terms
0:01:27.680,0:01:30.960
we can label them
0:01:28.560,0:01:32.400
H_x acting on phi where this is a
0:01:30.960,0:01:33.360
differential operator defined by this
0:01:32.400,0:01:36.640
equation
0:01:33.360,0:01:38.880
H_y of phi and H_z acting on phi
0:01:36.640,0:01:40.320
so separating the hamiltonian into
0:01:38.880,0:01:42.079
three parts
0:01:40.320,0:01:43.520
and then we can separate the energy also
0:01:42.079,0:01:45.759
in three parts which will define E:
0:01:43.520,0:01:47.520
E^x, E^y, and E^z but these are just
0:01:45.759,0:01:48.960
labels they're not raising to a power or
0:01:47.520,0:01:51.280
anything like that
0:01:48.960,0:01:52.479
so we can always do this of course if
0:01:51.280,0:01:54.560
we have some constant
0:01:52.479,0:01:55.840
energy we can just split that constant
0:01:54.560,0:01:58.640
term into three other
0:01:55.840,0:02:00.960
parts and ensure that the three add up
0:01:58.640,0:02:04.479
to the original value
0:02:00.960,0:02:05.600
so in doing this it makes it clear
0:02:04.479,0:02:06.240
that we're going to have a separable
0:02:05.600,0:02:07.439
equation
0:02:06.240,0:02:10.800
we can do it explicitly with the
0:02:07.439,0:02:14.480
following substitution
0:02:10.800,0:02:17.520
phi of x vector x is equal to
0:02:14.480,0:02:21.760
function capital X of x capital Y of y
0:02:17.520,0:02:25.280
capital Z of z if we substitute that in
0:02:21.760,0:02:26.640
we get the following result where
0:02:25.280,0:02:28.480
in each case we've pulled through the
0:02:26.640,0:02:31.599
two terms so y and z
0:02:28.480,0:02:34.160
which aren't acted on by the relevant
0:02:31.599,0:02:35.200
part of the hamiltonian so H of x is a
0:02:34.160,0:02:38.080
differential
0:02:35.200,0:02:39.920
operator which acts only on x and that
0:02:38.080,0:02:40.720
the capital x is the only function of x
0:02:39.920,0:02:42.879
and so on
0:02:40.720,0:02:45.280
we can divide through by x y z to get
0:02:42.879,0:02:46.720
the following result
0:02:45.280,0:02:48.800
where each of the terms now is a
0:02:46.720,0:02:50.560
function of only one of the variables
0:02:48.800,0:02:54.160
and on the right hand side we can use
0:02:50.560,0:02:57.519
our expression that we defined before
0:02:54.160,0:02:58.879
so the d squared by dx squared is now
0:02:57.519,0:03:01.440
our total derivative
0:02:58.879,0:03:03.680
because x is only a function of x this
0:03:01.440,0:03:04.959
is now an ordinary differential equation
0:03:03.680,0:03:07.599
and we know that the solutions take the
0:03:04.959,0:03:07.599
following form
0:03:07.840,0:03:12.080
that is the function x of x let's label
0:03:10.720,0:03:15.120
it with an integer
0:03:12.080,0:03:18.239
little and subscript x again x is just
0:03:15.120,0:03:21.599
a label here so nx is one of our
0:03:18.239,0:03:24.159
three integers and it's equal to this
0:03:21.599,0:03:25.519
properly normalized wave function
0:03:24.159,0:03:29.040
again here nx
0:03:25.519,0:03:31.920
pi x over l and the corresponding
0:03:29.040,0:03:32.319
eigen energy is E^x given the same
0:03:31.920,0:03:35.040
label
0:03:32.319,0:03:36.400
n_x and it takes the form of
0:03:35.040,0:03:37.920
the 1d result
0:03:36.400,0:03:40.159
so putting everything together we get
0:03:37.920,0:03:43.360
the same equations for y and z
0:03:40.159,0:03:44.560
and this is our total result so here's
0:03:43.360,0:03:48.080
our original equation
0:03:44.560,0:03:50.480
the solutions are phi of x multiplying
0:03:48.080,0:03:51.360
the the three separate solutions back
0:03:50.480,0:03:54.879
together
0:03:51.360,0:03:57.599
labeled by a vector of integers n
0:03:54.879,0:03:58.720
where n is defined by n=(nx,ny,nz)
0:03:58.720,0:04:01.920
the solutions to the three independent
0:04:00.640,0:04:02.879
equations each of these is just a
0:04:01.920,0:04:06.080
different integer
0:04:06.080,0:04:10.000
here's the product of the solutions in
0:04:07.599,0:04:13.519
the different directions and the energy
0:04:10.000,0:04:14.000
also labeled by vector n it takes
0:04:13.519,0:04:17.680
the usual
0:04:14.000,0:04:19.680
form so
0:04:17.680,0:04:20.880
we can define if we think back to our
0:04:19.680,0:04:23.840
quantum numbers
0:04:20.880,0:04:25.520
remember that's some quantity
0:04:23.840,0:04:28.160
corresponding to an expectation value
0:04:25.520,0:04:30.720
which doesn't vary with time
0:04:28.160,0:04:32.479
if we look at our three separate
0:04:30.720,0:04:35.520
hamiltonians that we separated the
0:04:32.479,0:04:38.080
original hamiltonian into
0:04:35.520,0:04:39.759
we have the sum of three terms where any
0:04:38.080,0:04:41.280
two of these terms commute because
0:04:39.759,0:04:43.919
each one is only a function of one of
0:04:41.280,0:04:47.199
the different variables
0:04:43.919,0:04:49.360
and so what this tells us is that
0:04:47.199,0:04:52.240
we can define the quantum numbers for
0:04:49.360,0:04:52.240
each of these different
0:04:53.280,0:04:57.520
parts of the hamiltonian we can define
0:04:56.160,0:04:59.919
these three quantum numbers
0:04:57.520,0:05:00.639
at the same time so we can write states
0:04:59.919,0:05:02.720
that
0:05:00.639,0:05:03.759
are labeled by all three of these and
0:05:02.720,0:05:05.600
there's no
0:05:03.759,0:05:07.600
problem with specifying them we can have
0:05:05.600,0:05:08.639
simultaneous knowledge of nx, ny and
0:05:07.600,0:05:10.880
nz
0:05:08.639,0:05:12.960
so in ket notation it's convenient to
0:05:10.880,0:05:16.160
define the following
0:05:12.960,0:05:18.880
that is our wave function phi n of x
0:05:16.160,0:05:19.840
is the x projection where x is now
0:05:18.880,0:05:21.840
the vector
0:05:19.840,0:05:24.000
but still projection into three
0:05:21.840,0:05:26.320
dimensional position space now
0:05:24.000,0:05:27.840
of some ket |nx,ny,nz>
0:05:27.840,0:05:32.400
where we have the following time
0:05:29.199,0:05:32.400
independent schrodinger equation
0:05:32.720,0:05:38.960
with E_n defined above
0:05:36.160,0:05:40.880
so the ground state of the system for
0:05:38.960,0:05:42.560
example let's clear the board
0:05:40.880,0:05:44.560
the ground state of the system is the
0:05:42.560,0:05:46.960
state
0:05:44.560,0:05:48.639
labeled with a ket
0:05:46.960,0:05:50.639
|1,1,1>
0:05:48.639,0:05:52.720
so all of the ns are as low as
0:05:50.639,0:05:55.840
possible and the corresponding
0:05:52.720,0:05:59.199
energy eigenvalue is
0:05:55.840,0:06:01.039
E_{1,1,1} we substitute
0:05:59.199,0:06:02.080
the ones in and we find hbar squared pi
0:06:01.039,0:06:05.120
squared over two
0:06:02.080,0:06:07.600
m l squared multiplied by three and
0:06:05.120,0:06:09.440
the ground state is unique
0:06:09.440,0:06:13.360
there there's only one state with
0:06:11.919,0:06:16.000
that energy
0:06:13.360,0:06:16.880
however the first excited state there
0:06:16.000,0:06:18.560
are three
0:06:16.880,0:06:20.800
independent states which all have the
0:06:18.560,0:06:24.639
same energy
0:06:20.800,0:06:25.120
so these are the the states with the
0:06:24.639,0:06:28.720
next
0:06:25.120,0:06:32.000
lowest energy and all three of these
0:06:28.720,0:06:33.199
have the same energy which is stated
0:06:32.000,0:06:35.440
here
0:06:33.199,0:06:36.240
so now we start to have degenerate
0:06:35.440,0:06:38.400
states
0:06:36.240,0:06:39.600
multiple different states with the
0:06:38.400,0:06:41.280
same energies
0:06:39.600,0:06:43.039
so some of the results we've seen before
0:06:41.280,0:06:45.520
relied on having
0:06:43.039,0:06:47.440
non-degenerate eigenvalues some of those
0:06:45.520,0:06:50.400
will disappear but in fact
0:06:47.440,0:06:52.560
it's still possible to find an
0:06:50.400,0:06:56.400
orthonormal basis for these states
0:06:52.560,0:06:59.120
in the following form so for our vector
0:06:56.400,0:06:59.840
for our state n x and y and z if we take
0:06:59.120,0:07:02.080
another state
0:06:59.840,0:07:04.240
|m_x,m_y,m_z> with these different
0:07:02.080,0:07:05.919
integers or potentially the same
0:07:04.240,0:07:07.759
we find that the inner product of these
0:07:05.919,0:07:09.599
two states is a product
0:07:07.759,0:07:13.680
of the Kronecker deltas for each of the
0:07:09.599,0:07:15.120
three pairs so mx has to equal nx
0:07:13.680,0:07:18.000
otherwise this is zero and the whole
0:07:15.120,0:07:21.280
thing is zero so we need mx equals nx
0:07:18.000,0:07:22.880
my equals ny and mz equals nz
0:07:21.280,0:07:24.400
and this is quite straight forward to
0:07:22.880,0:07:26.160
prove as usual by
0:07:24.400,0:07:28.400
insertion of the identity which takes
0:07:26.160,0:07:29.840
the following form
0:07:28.400,0:07:32.400
that is we've just taken this inner
0:07:29.840,0:07:34.080
product here and inserted a complete
0:07:32.400,0:07:35.280
set of position states where again these
0:07:34.080,0:07:37.199
are three dimensional now
0:07:35.280,0:07:38.479
and so our volume integral must be
0:07:37.199,0:07:40.880
dx dy dz
0:07:38.479,0:07:42.000
but we know what this is from the
0:07:40.880,0:07:44.000
previous board
0:07:42.000,0:07:45.919
it's just the the wave function we
0:07:44.000,0:07:47.199
found before and therefore this must be
0:07:45.919,0:07:50.479
its complex conjugate
0:07:47.199,0:07:53.599
that is these two things read
0:07:50.479,0:07:56.639
so sine mx pi x over l
0:07:53.599,0:07:58.960
same for y and z and then the same for
0:07:56.639,0:07:59.759
the ns but we can just separate these
0:07:58.960,0:08:02.080
into three
0:07:59.759,0:08:02.879
separate integrals each of which
0:08:02.080,0:08:06.240
multiplies
0:08:02.879,0:08:06.879
the others so the first is just two over
0:08:06.240,0:08:10.160
l
0:08:06.879,0:08:13.599
integral over zero to l of dx
0:08:10.160,0:08:16.720
of the sine of mx pi x over l
0:08:13.599,0:08:17.280
sine nx pi x over l and you get the
0:08:16.720,0:08:19.919
same
0:08:17.280,0:08:20.960
with x change to y and then x change to
0:08:19.919,0:08:22.960
z
0:08:20.960,0:08:24.960
but this is nothing other than the
0:08:22.960,0:08:26.240
integral we used in the one dimensional
0:08:24.960,0:08:29.680
infinite potential well
0:08:26.240,0:08:29.680
which equals our kronecker delta
0:08:29.919,0:08:33.120
so this is nothing other than the
0:08:32.159,0:08:35.599
kronecker delta
0:08:33.120,0:08:36.880
telling us that mx has to equal nx you
0:08:35.599,0:08:39.360
get the same for the y
0:08:36.880,0:08:41.120
the y and z terms and this is how
0:08:39.360,0:08:42.640
you prove this result up here
0:08:41.120,0:08:44.320
so in this case actually we have to
0:08:42.640,0:08:44.880
generate states which is potentially a
0:08:44.320,0:08:48.080
problem
0:08:44.880,0:08:50.000
but actually it's possible to choose
0:08:48.080,0:08:51.760
an orthonormal basis just like we did in
0:08:50.000,0:08:52.720
the 1d case it's not really much more
0:08:51.760,0:08:55.839
difficult
0:08:52.720,0:08:55.839
okay thank you for your time
V9.2 The 3D harmonic oscillator
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
the harmonic oscillator in three dimensions: eigenstates and eigenvalues; separating into three independent one-dimensional problems.
0:00:00.399,0:00:04.160
hello in this short video we're going to
0:00:02.800,0:00:06.080
take a look at the three-dimensional
0:00:04.160,0:00:08.080
quantum harmonic oscillator
0:00:06.080,0:00:09.519
in fact the moral of the story is
0:00:08.080,0:00:10.960
that it's really no more difficult than
0:00:09.519,0:00:12.160
the one-dimensional quantum harmonic
0:00:10.960,0:00:14.559
oscillator
0:00:12.160,0:00:16.160
and i'm going to assume familiarity
0:00:14.559,0:00:17.600
with the results of that case
0:00:16.160,0:00:19.600
and work quite quickly through this
0:00:17.600,0:00:21.680
example so the potential in this case is
0:00:19.600,0:00:24.880
as follows
0:00:21.680,0:00:26.720
which equals half m omega squared x
0:00:24.880,0:00:29.599
operator squared plus y operator squared
0:00:26.720,0:00:31.279
plus z operator squared
0:00:29.599,0:00:32.719
the most convenient way to solve it is
0:00:31.279,0:00:34.079
again using raising and lowering
0:00:32.719,0:00:35.440
operators but this time you need to
0:00:34.079,0:00:37.040
define a different
0:00:35.440,0:00:38.879
raising and lowering operator for
0:00:37.040,0:00:39.920
each of the three directions which are
0:00:38.879,0:00:43.200
perpendicular
0:00:39.920,0:00:44.320
so we have the raising operator in the x
0:00:43.200,0:00:46.399
direction
0:00:44.320,0:00:48.239
is given by this expression square
0:00:46.399,0:00:51.039
root of omega over two h bar
0:00:48.239,0:00:51.920
multiplying the x operator minus i over
0:00:51.039,0:00:54.719
m omega
0:00:51.920,0:00:57.120
p x where in the position basis this
0:00:54.719,0:01:00.399
is as follows
0:00:57.120,0:01:01.520
where partial subscript x again as
0:01:00.399,0:01:04.799
usual means
0:01:01.520,0:01:08.400
d by dx while holding y z
0:01:04.799,0:01:10.240
and in this case time all constant
0:01:08.400,0:01:11.680
so we have the same raising and
0:01:10.240,0:01:12.000
lowering operators and we have the same
0:01:11.680,0:01:13.520
for
0:01:12.000,0:01:15.040
what the et cetera here means same
0:01:13.520,0:01:16.799
for y and z
0:01:15.040,0:01:18.960
we can find the number operator in each
0:01:16.799,0:01:22.400
direction
0:01:18.960,0:01:25.280
and so we can rewrite the hamiltonian
0:01:22.400,0:01:26.080
as the sum of three terms the nx
0:01:25.280,0:01:28.000
operator
0:01:26.080,0:01:30.079
plus the ny operator and z
0:01:28.000,0:01:32.320
operator and then we get plus
0:01:30.079,0:01:34.159
one half times the identity operator for
0:01:32.320,0:01:36.799
each of these so plus three over two
0:01:34.159,0:01:41.439
identity operator in total so the
0:01:36.799,0:01:43.759
time independent Schrodinger equation reads
0:01:41.439,0:01:45.119
that is the number operator acting on
0:01:43.759,0:01:48.399
the respective
0:01:45.119,0:01:50.479
state returns the integer
0:01:48.399,0:01:53.360
number in exactly the same way that it
0:01:50.479,0:01:55.280
would in the one-dimensional case
0:01:53.360,0:01:56.960
the fact that we've been able to
0:01:55.280,0:01:59.840
write states like this
0:01:56.960,0:02:01.439
labeled by nx ny and nz means that
0:01:59.840,0:02:03.280
nx ny and nz must
0:02:01.439,0:02:04.799
individually be good quantum numbers that
0:02:03.280,0:02:05.280
is we can define all three at the same
0:02:04.799,0:02:08.959
time
0:02:05.280,0:02:11.280
we can have simultaneous knowledge and
0:02:08.959,0:02:12.879
just as we saw in the three-dimensional
0:02:11.280,0:02:14.800
infinite potential well this is
0:02:12.879,0:02:16.640
true the equation is once again
0:02:14.800,0:02:20.879
separable where we can
0:02:16.640,0:02:22.640
write the following for the hamiltonian
0:02:20.879,0:02:24.560
that is the hamiltonian can be written
0:02:22.640,0:02:27.920
as the sum of three terms where
0:02:24.560,0:02:28.879
this term only acts on the x
0:02:27.920,0:02:30.879
direction
0:02:28.879,0:02:32.720
so the variable x this one only acts on
0:02:30.879,0:02:35.760
y and this one only acts on z
0:02:32.720,0:02:37.760
and for example the Hx operator is the
0:02:35.760,0:02:38.480
momentum in the x direction squared over
0:02:37.760,0:02:41.920
2m
0:02:38.480,0:02:44.959
where this is defined up here plus
0:02:41.920,0:02:47.120
half m omega squared x operator squared
0:02:44.959,0:02:48.080
and then the et cetera indicates that we
0:02:47.120,0:02:49.680
get the same for y
0:02:48.080,0:02:52.080
and z but this would then be the y
0:02:49.680,0:02:56.959
operations that operate
0:02:52.080,0:02:56.959
okay thank you for your time
V9.3 Angular momentum
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
angular momentum in quantum mechanics: operators, commutations relations. The squared total angular momentum operator and its commutation with the other operators; forming a maximal set of commuting operators. Spherical polar co-ordinates.
0:00:00.080,0:00:04.720
hello this week we're going to talk
0:00:01.680,0:00:04.720
about angular momentum
0:00:11.440,0:00:17.039
so classically the angular momentum is
0:00:13.200,0:00:19.039
defined as follows
0:00:17.039,0:00:20.640
that is L the three dimensional angular
0:00:19.039,0:00:23.519
momentum is defined by
0:00:20.640,0:00:24.400
x the three dimensional vector x y z
0:00:23.519,0:00:28.240
giving the
0:00:24.400,0:00:31.359
direction cross the momentum
0:00:28.240,0:00:32.719
p again px py pz so
0:00:31.359,0:00:34.320
in quantum mechanics our angular
0:00:32.719,0:00:34.800
momentum operator is exactly the same
0:00:34.320,0:00:37.920
thing
0:00:34.800,0:00:40.239
it's just that all classical
0:00:37.920,0:00:41.360
observables get promoted to operators in
0:00:40.239,0:00:44.480
quantum mechanics
0:00:41.360,0:00:45.039
so we have the following that is L is
0:00:44.480,0:00:48.879
still
0:00:45.039,0:00:51.600
x cross p but x and p are now
0:00:48.879,0:00:52.800
vectors of operators and and
0:00:51.600,0:00:54.640
this is a general rule in quantum
0:00:52.800,0:00:56.960
mechanics
0:00:54.640,0:00:58.320
classical observables become operators
0:00:56.960,0:01:00.079
in quantum mechanics
0:00:58.320,0:01:02.559
and this is a straightforward way to
0:01:00.079,0:01:04.479
form the the quantum operator it'll
0:01:02.559,0:01:06.320
give us the angular momentum
0:01:04.479,0:01:07.600
so it takes the following form if we
0:01:06.320,0:01:11.439
multiply out the
0:01:07.600,0:01:12.320
cross product that is sorry the operator
0:01:11.439,0:01:15.600
L
0:01:12.320,0:01:18.720
vector is minus i h bar x
0:01:15.600,0:01:21.840
cross grad whereas now we're
0:01:18.720,0:01:23.520
working in the position basis
0:01:21.840,0:01:25.840
and we're also working in cartesian
0:01:23.520,0:01:28.960
coordinates
0:01:25.840,0:01:29.520
so we can multiply out x cross
0:01:28.960,0:01:31.360
grad
0:01:29.520,0:01:33.200
and we get the following result so we
0:01:31.360,0:01:35.600
have a minus i hbar out the front
0:01:33.200,0:01:37.360
and then we have for example y then
0:01:35.600,0:01:40.960
partial derivative d by
0:01:37.360,0:01:43.040
d z holding x and y and time
0:01:40.960,0:01:45.520
constant and so on for all these
0:01:43.040,0:01:48.799
different terms
0:01:45.520,0:01:52.560
and we can re-express this as the vector
0:01:48.799,0:01:54.240
L=(Lx,Ly,Lz) where each of these is the
0:01:52.560,0:01:56.399
angular momentum operator in each of the
0:01:54.240,0:01:58.320
three different cartesian directions
0:01:56.399,0:01:59.840
and the corresponding terms are given
0:01:58.320,0:02:01.439
over here
0:01:59.840,0:02:02.880
so we can work with these things as
0:02:01.439,0:02:03.680
operators and look at their commutation
0:02:02.880,0:02:06.799
relations
0:02:03.680,0:02:07.040
and we find the following the commutator
0:02:06.799,0:02:11.039
of
0:02:07.040,0:02:15.760
lx with ly is given by i h bar
0:02:11.039,0:02:18.560
l z and etc here means that we can
0:02:15.760,0:02:19.120
do the do the cyclic permutations to
0:02:18.560,0:02:22.959
get
0:02:19.120,0:02:26.319
for example [Ly,Lz] is i h bar
0:02:22.959,0:02:29.680
Lx and so on so
0:02:26.319,0:02:33.280
they don't commute we can't have
0:02:29.680,0:02:35.360
mutual knowledge of all three
0:02:33.280,0:02:37.200
angular momenta in all the three
0:02:35.360,0:02:38.959
different cartesian directions
0:02:37.200,0:02:40.720
it's only possible to have
0:02:38.959,0:02:43.519
information about one of them
0:02:40.720,0:02:45.040
at a given time so notice that this is
0:02:43.519,0:02:47.040
actually exactly the same structure that
0:02:45.040,0:02:50.319
we saw when we looked at spin
0:02:47.040,0:02:52.879
back in an earlier video so
0:02:50.319,0:02:54.800
spin if you may recall is intrinsic
0:02:52.879,0:02:56.319
angular momentum and so it's perhaps
0:02:54.800,0:02:57.920
not unreasonable to expect that we
0:02:56.319,0:03:01.599
get exactly the same structure
0:02:57.920,0:03:04.239
for the usual angular momentum
0:03:01.599,0:03:05.840
so here's our commutation relation again
0:03:04.239,0:03:06.480
so this means we can't have simultaneous
0:03:05.840,0:03:09.840
knowledge of
0:03:06.480,0:03:11.280
Lx, Ly, and Lz
0:03:09.840,0:03:12.959
in fact we can quantify this with the
0:03:11.280,0:03:14.640
heisenberg uncertainty principle
0:03:12.959,0:03:15.040
remember that for general operators a
0:03:14.640,0:03:18.080
and b
0:03:15.040,0:03:20.080
this states that the uncertainty in a
0:03:18.080,0:03:20.800
defined by the standard deviation of the
0:03:20.080,0:03:22.720
operator
0:03:20.800,0:03:24.000
multiplied by the uncertainty of b is
0:03:22.720,0:03:27.040
greater than or equal to
0:03:24.000,0:03:29.920
half of the modulus of the
0:03:27.040,0:03:30.720
expectation value of the commutator of a
0:03:29.920,0:03:33.840
with b
0:03:30.720,0:03:33.840
so in this case we find the result
0:03:34.000,0:03:38.239
the product of the uncertainties is
0:03:35.440,0:03:39.519
greater than or equal to h bar over 2
0:03:38.239,0:03:41.760
multiplying the magnitude of the
0:03:39.519,0:03:44.720
expectation value of the third operator
0:03:41.760,0:03:45.760
so it's not zero actually there is an
0:03:44.720,0:03:49.280
exceptional case
0:03:45.760,0:03:50.480
in which you can know Lx, Ly, and Lz
0:03:49.280,0:03:51.440
and that's when all three are equal to
0:03:50.480,0:03:53.760
zero
0:03:51.440,0:03:55.840
which this expression accounts for
0:03:53.760,0:03:58.319
but other than that kind of boring case
0:03:55.840,0:04:00.000
you can simultaneously
0:03:58.319,0:04:01.920
only know one of the three
0:04:00.000,0:04:04.080
however we can define the following
0:04:01.920,0:04:06.799
operator
0:04:04.080,0:04:09.040
the square of the total angular momentum
0:04:06.799,0:04:10.319
L^2 is
0:04:09.040,0:04:11.519
Lx^2+Ly^2+Lz^2
0:04:10.319,0:04:13.360
and actually it's quite straightforward
0:04:11.519,0:04:14.000
to check that this commutes with all
0:04:13.360,0:04:16.479
three of
0:04:14.000,0:04:18.239
Lx, Ly, and Lz for example let's check
0:04:16.479,0:04:20.320
with lx
0:04:18.239,0:04:22.160
the comutator of lx with Ly squared
0:04:20.320,0:04:22.800
plus the commutator of lx with lz
0:04:22.160,0:04:24.560
squared
0:04:22.800,0:04:27.120
because the commutator of lx with lx
0:04:24.560,0:04:29.440
squared is zero
0:04:27.120,0:04:31.759
for each of these two we can expand them
0:04:29.440,0:04:34.000
using the relation
0:04:34.000,0:04:37.919
[a,b^2]
0:04:35.120,0:04:46.160
is [a,b]b + b[a,b]
0:04:43.040,0:04:48.160
and so we get the following so the sum
0:04:46.160,0:04:49.840
of these four terms
0:04:48.160,0:04:51.280
and sticking in for example lx
0:04:49.840,0:04:54.000
commutator ly
0:04:51.280,0:04:55.680
is i h bar lz we get the following
0:04:54.000,0:04:58.880
result
0:04:55.680,0:04:59.919
these four terms were the first term i h
0:04:58.880,0:05:03.680
bar lz; ly
0:04:59.919,0:05:06.720
cancels with minus i h bar
0:05:03.680,0:05:09.120
lz ly and the second term
0:05:06.720,0:05:09.840
cancels with this term and we find that
0:05:09.120,0:05:12.639
lx
0:05:09.840,0:05:13.840
commutator L^2 is indeed zero so
0:05:12.639,0:05:17.919
what this tells us
0:05:13.840,0:05:20.479
is as follows so to recap we have
0:05:17.919,0:05:21.759
the commutator of any of the L_i
0:05:20.479,0:05:25.360
where i equals
0:05:21.759,0:05:27.680
x y or z with L squared is zero
0:05:25.360,0:05:28.639
but the commutator of any L_i with L_j
0:05:27.680,0:05:31.600
where both i and j
0:05:28.639,0:05:32.560
are either x y or z is not equal to
0:05:31.600,0:05:35.840
zero unless
0:05:32.560,0:05:39.280
for any i does not equal j so
0:05:35.840,0:05:40.479
l x comma l y here is is not equal to
0:05:39.280,0:05:41.840
zero for example
0:05:40.479,0:05:44.560
so what this tells us is that we can't
0:05:41.840,0:05:47.840
have simultaneous knowledge of all three
0:05:44.560,0:05:49.120
components of the angular momentum
0:05:47.840,0:05:51.120
we can only have knowledge of one of
0:05:49.120,0:05:52.960
those at a time but we can have
0:05:51.120,0:05:55.919
knowledge of any one of those
0:05:52.960,0:05:57.440
and l squared the square of the total
0:05:55.919,0:05:58.000
angular momentum you can think of this
0:05:57.440,0:05:59.600
because
0:05:58.000,0:06:01.840
classically this would just be the
0:05:59.600,0:06:06.160
length of the angular momentum
0:06:01.840,0:06:09.600
vector
0:06:06.160,0:06:11.440
and so we can define our angular
0:06:09.600,0:06:12.720
momentum states with two
0:06:11.440,0:06:14.720
quantum numbers that can be known
0:06:12.720,0:06:18.720
simultaneously one corresponding
0:06:14.720,0:06:20.880
to any choice of lx ly or lz
0:06:18.720,0:06:22.880
and the other corresponding to the
0:06:20.880,0:06:25.520
square of the total angular momentum
0:06:22.880,0:06:26.240
so we can write the following we can
0:06:25.520,0:06:29.360
define
0:06:26.240,0:06:31.039
a state |l,m> where l and m are
0:06:29.360,0:06:33.680
integers
0:06:31.039,0:06:35.440
such that the operator L squared acting
0:06:33.680,0:06:36.720
on this state returns the corresponding
0:06:35.440,0:06:38.639
eigenvalue
0:06:36.720,0:06:40.880
now we'll see when we look in more
0:06:38.639,0:06:41.600
detail at this in a couple of videos'
0:06:40.880,0:06:44.400
time
0:06:41.600,0:06:46.000
that the form of the eigenvalue is most
0:06:44.400,0:06:49.599
convenient to define it in this way
0:06:46.000,0:06:52.319
h bar squared l(l+1)
0:06:49.599,0:06:55.039
where l is an integer here which is
0:06:52.319,0:06:57.199
equal to or greater than zero
0:06:55.039,0:06:58.160
acting again on the state and
0:06:57.199,0:07:00.319
similarly
0:06:58.160,0:07:02.160
choosing for convenience the lz
0:07:00.319,0:07:02.639
component we have to pick one of the
0:07:02.160,0:07:05.360
three
0:07:02.639,0:07:05.919
and lz is particularly convenient as
0:07:05.360,0:07:07.440
it's the
0:07:05.919,0:07:08.880
simplest when we write them in
0:07:07.440,0:07:10.240
spherical polar coordinates which we'll
0:07:08.880,0:07:13.440
do in a second
0:07:10.240,0:07:14.080
so we get the following so we can
0:07:13.440,0:07:16.319
define
0:07:14.080,0:07:17.360
exactly the same state |l,m> and
0:07:16.319,0:07:19.440
it's also got
0:07:17.360,0:07:21.120
a simultaneous
0:07:19.440,0:07:23.199
eigenstate of Lz
0:07:21.120,0:07:24.800
and l squared at the same time and the
0:07:23.199,0:07:27.919
eigenvalue of lz
0:07:24.800,0:07:29.599
we define as h bar m where m here is
0:07:27.919,0:07:31.680
an integer
0:07:29.599,0:07:33.199
which it's unfortunate that it's called
0:07:31.680,0:07:33.919
m because of course we use m for the
0:07:33.199,0:07:35.840
mass
0:07:33.919,0:07:37.520
but this is the convention that tends
0:07:35.840,0:07:39.680
to be used so hopefully
0:07:37.520,0:07:40.880
it's not too confusing that m here is
0:07:39.680,0:07:43.919
an integer
0:07:40.880,0:07:45.280
okay and sorry when I said m is an
0:07:43.919,0:07:47.280
integer on that previous board
0:07:45.280,0:07:49.199
m is truly just unrelated to the mass
0:07:47.280,0:07:51.360
it's just an
0:07:49.199,0:07:52.560
integer which you happen to label m.
0:07:51.360,0:07:54.560
We'll see much more of it
0:07:52.560,0:07:57.440
in the coming videos okay so let's look
0:07:54.560,0:08:00.160
at spherical polar coordinates
0:07:57.440,0:08:01.759
so still in the position
0:08:00.160,0:08:02.720
basis as we were in the cartesian
0:08:01.759,0:08:04.639
coordinates
0:08:02.720,0:08:06.240
we can define spherical polar
0:08:04.639,0:08:08.000
coordinates are theta and phi in the
0:08:06.240,0:08:12.240
usual manner
0:08:08.000,0:08:15.280
so an azimuthal angle coming down
0:08:12.240,0:08:18.800
from z the
0:08:15.280,0:08:22.319
radial distance and
0:08:18.800,0:08:25.280
the polar angle phi
0:08:22.319,0:08:27.120
all in the usual form then if we just
0:08:25.280,0:08:29.440
write out the components of the angular
0:08:27.120,0:08:32.080
momentum
0:08:29.440,0:08:33.680
defined as before as x cross p or
0:08:32.080,0:08:37.039
minus i h bar x
0:08:33.680,0:08:38.479
cross grad we find
0:08:37.039,0:08:40.000
I'm only going to write the l z
0:08:38.479,0:08:41.919
component because it's the most the
0:08:40.000,0:08:44.080
simplest to write
0:08:41.919,0:08:45.519
that is in this position basis in
0:08:44.080,0:08:48.240
spherical polar coordinates
0:08:45.519,0:08:48.959
the lz operator is minus i h bar d by d
0:08:48.240,0:08:54.720
phi
0:08:48.959,0:08:57.200
holding r and theta constant and time
0:08:54.720,0:08:59.600
lx and ly are a bit more complicated but
0:08:57.200,0:09:01.200
it's not too hard to work them out
0:08:59.600,0:09:02.640
and it's also convenient to write down
0:09:01.200,0:09:05.760
the l squared
0:09:02.640,0:09:08.560
angular momentum operator
0:09:05.760,0:09:09.440
which comes out as the following and
0:09:09.440,0:09:13.600
the partial derivative subscript
0:09:11.920,0:09:15.600
notation is coming into its own here as
0:09:13.600,0:09:19.120
things are getting a little bit
0:09:15.600,0:09:19.680
painful to write down it's worth
0:09:19.120,0:09:22.240
noting
0:09:19.680,0:09:24.080
that the units
0:09:22.240,0:09:26.959
of angular momentum are actually exactly
0:09:24.080,0:09:28.959
the same as the units of h bar
0:09:26.959,0:09:31.040
and so it's easy to check whether we've
0:09:28.959,0:09:32.240
got the right power of h bar in our
0:09:31.040,0:09:34.560
expression
0:09:32.240,0:09:36.560
and another thing worth noting is
0:09:34.560,0:09:39.200
that the hamiltonian
0:09:36.560,0:09:40.240
can be written as follows so the
0:09:39.200,0:09:42.880
hamiltonian
0:09:40.240,0:09:44.160
in the position basis in spherical
0:09:42.880,0:09:47.279
polar coordinates
0:09:44.160,0:09:49.519
acting on psi(x,t) where I'm
0:09:47.279,0:09:51.680
really writing psi rather than the usual
0:09:49.519,0:09:53.600
phi(x) just because phi is being
0:09:51.680,0:09:57.040
used now as our
0:09:53.600,0:10:00.800
polar coordinate is equal to minus h bar
0:09:57.040,0:10:03.600
squared over 2 m grad squared plus
0:10:00.800,0:10:04.480
V(x) acting on psi(x,t) and
0:10:03.600,0:10:06.640
expanding the grad
0:10:04.480,0:10:08.800
squared is the thing that differs in
0:10:06.640,0:10:11.200
this case from cartesians
0:10:08.800,0:10:11.839
we get something which acts only on the
0:10:11.200,0:10:16.079
coordinate
0:10:11.839,0:10:16.800
r plus the l squared operator over 2m r
0:10:16.079,0:10:18.720
squared
0:10:16.800,0:10:20.079
plus the potential written in spherical
0:10:18.720,0:10:24.079
polar coordinates all
0:10:20.079,0:10:27.519
acting on psi(x,t)
0:10:24.079,0:10:28.880
so having knowledge of this
0:10:27.519,0:10:29.920
L squared operator is particularly
0:10:28.880,0:10:31.200
convenient we'll see
0:10:29.920,0:10:33.360
much more of this when we come back to
0:10:31.200,0:10:38.079
study the hydrogen atom shortly
0:10:33.360,0:10:38.079
thank you for your time
V9.4 Angular momentum ladder operators
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
defining raising and lowering (ladder) operators for the z-projection of angular momentum.
0:00:00.320,0:00:04.160
hello in this video we're going to take
0:00:02.320,0:00:04.960
a look at angular momentum again but
0:00:04.160,0:00:07.440
this time
0:00:04.960,0:00:08.000
using ladder operators quite a neat
0:00:07.440,0:00:09.519
trick
0:00:08.000,0:00:11.440
for evaluating some of the properties
0:00:09.519,0:00:12.480
that we derived in length in a previous
0:00:11.440,0:00:15.440
video
0:00:12.480,0:00:16.720
so we've seen that the commutator of
0:00:15.440,0:00:18.160
angular momentum operates as
0:00:16.720,0:00:21.119
follows
0:00:18.160,0:00:21.840
[Lx,Ly] is equal to y
0:00:21.119,0:00:24.400
h bar Lz
0:00:21.840,0:00:26.320
and the etc here just indicates that
0:00:24.400,0:00:29.760
you can take cyclic permutations
0:00:26.320,0:00:32.960
of x y and z to arrive at equivalent
0:00:29.760,0:00:34.640
results for the other two cases
0:00:32.960,0:00:37.200
and it's convenient to define the
0:00:34.640,0:00:40.559
following operators
0:00:37.200,0:00:43.600
L_{plus/minus} equal by definition to
0:00:40.559,0:00:45.360
lx plus or minus i ly so
0:00:43.600,0:00:46.640
if we look at these operators and their
0:00:45.360,0:00:49.200
commutators
0:00:46.640,0:00:50.640
with the remaining terms we find
0:00:49.200,0:00:53.600
the following
0:00:50.640,0:00:54.800
so l plus or minus with lz commutator
0:00:53.600,0:00:57.520
if we can expand it
0:00:54.800,0:00:58.239
lx commutes with lz plus or minus i
0:00:57.520,0:01:01.440
times
0:00:58.239,0:01:04.239
l y commutes with lz
0:01:01.440,0:01:06.240
we can substitute the various forms of
0:01:04.239,0:01:08.880
this expression into these two places to
0:01:06.240,0:01:11.600
find the result
0:01:08.880,0:01:14.320
so we find this but this can be factored
0:01:11.600,0:01:14.320
into the form
0:01:14.400,0:01:19.439
where we've got the l plus or minus
0:01:16.320,0:01:21.600
back again and so the final result
0:01:19.439,0:01:23.759
the commutator of l plus or minus with l
0:01:21.600,0:01:28.880
z is equal to minus plus
0:01:23.759,0:01:31.520
h bar l plus or minus let's box that
0:01:28.880,0:01:32.000
so the reason this is convenient is that
0:01:31.520,0:01:34.400
we've
0:01:32.000,0:01:35.759
seen previously that we can define
0:01:34.400,0:01:39.119
eigenstates of the lz
0:01:35.759,0:01:39.119
operator as follows
0:01:39.280,0:01:44.960
so lz acting on state m where m is an
0:01:42.560,0:01:46.640
integer m is not related to the mass at
0:01:44.960,0:01:48.240
all it's just an integer and this is a
0:01:46.640,0:01:49.759
conventional label which is a little bit
0:01:48.240,0:01:51.600
unfortunate because of the confusion
0:01:49.759,0:01:54.240
with mass
0:01:51.600,0:01:56.240
is equal to h bar m same integer
0:01:54.240,0:01:57.600
multiplying the state m
0:01:56.240,0:01:59.439
so let's look at what happens when we
0:01:57.600,0:02:01.680
act on both sides of this with l plus or
0:01:59.439,0:02:03.280
minus
0:02:01.680,0:02:05.360
so on the left hand side we have l plus
0:02:03.280,0:02:06.159
or minus acting on lz acting on m as
0:02:05.360,0:02:07.600
before
0:02:06.159,0:02:09.200
then we have on the right hand side h
0:02:07.600,0:02:10.720
but the l plus or minus comes through
0:02:09.200,0:02:12.640
and acts on the state
0:02:10.720,0:02:14.239
so over here we'd like to commute the lz
0:02:12.640,0:02:15.440
through the l plus or minus
0:02:14.239,0:02:16.800
and i should say that when i'm writing l
0:02:15.440,0:02:18.879
plus or minus of course this is two
0:02:16.800,0:02:20.640
separate operators one is l plus with a
0:02:18.879,0:02:22.160
plus sign here and one is l minus with a
0:02:20.640,0:02:23.760
minus sign here we're just
0:02:22.160,0:02:26.400
saving ourselves some time by doing both
0:02:23.760,0:02:28.640
at once with a convenient notation
0:02:26.400,0:02:31.280
so we can pull these the lz through the
0:02:28.640,0:02:33.280
l plus or minus as follows
0:02:31.280,0:02:34.400
that is we can write lz to the left of l
0:02:33.280,0:02:36.000
plus or minus
0:02:34.400,0:02:37.680
added to the commutator of l plus or
0:02:36.000,0:02:40.560
minus with lz because the
0:02:37.680,0:02:42.239
the second term of this is minus lz
0:02:40.560,0:02:44.080
l plus minus which cancels this one
0:02:42.239,0:02:46.160
the first term is l plus or minus l z
0:02:44.080,0:02:49.440
which is the thing we started with
0:02:46.160,0:02:52.720
this commutator we've just found
0:02:49.440,0:02:54.560
as follows we can take this term over to
0:02:52.720,0:02:57.599
the other side
0:02:54.560,0:03:00.720
to get the end result
0:02:57.599,0:03:02.640
so lz acting on the
0:03:00.720,0:03:04.000
state inside the parentheses here is
0:03:02.640,0:03:06.400
equal to h bar
0:03:04.000,0:03:08.000
m plus or minus 1 acting on the same
0:03:06.400,0:03:10.800
state inside the parentheses
0:03:08.000,0:03:11.680
and so m by definition was an eigenstate
0:03:10.800,0:03:15.440
m
0:03:11.680,0:03:17.680
l z with eigenvalue h bar m
0:03:15.440,0:03:20.159
and so l plus or minus acting on the
0:03:17.680,0:03:22.560
state m must also be an eigen state
0:03:20.159,0:03:23.840
because just ignoring what's written
0:03:22.560,0:03:25.519
inside the parentheses
0:03:23.840,0:03:27.760
this remaining thing is an eigenvalue
0:03:25.519,0:03:30.000
equation for lz
0:03:27.760,0:03:32.319
and this must also be an eigenstate and
0:03:30.000,0:03:33.280
the eigenvalue must be h bar m plus or
0:03:32.319,0:03:36.000
minus one
0:03:33.280,0:03:36.879
so you see that l plus acting on state m
0:03:36.000,0:03:40.000
raises
0:03:36.879,0:03:43.519
the eigenvalue by one or
0:03:40.000,0:03:46.239
h bar times one and l minus
0:03:43.519,0:03:48.239
lowers it again and so these l plus or
0:03:46.239,0:03:49.920
minus are angular momentum
0:03:48.239,0:03:53.680
raising and lowering operators or
0:03:49.920,0:03:53.680
angular momentum ladder operators
0:03:54.640,0:04:02.159
so if you recall from
0:03:57.840,0:04:03.439
the previous video that l x l y and l z
0:04:02.159,0:04:04.080
while they don't commute with one
0:04:03.439,0:04:06.560
another
0:04:04.080,0:04:09.040
they all commute with l squared,
0:04:06.560,0:04:12.640
the square of the angular momentum operator
0:04:09.040,0:04:15.760
and so that fact tells us that
0:04:12.640,0:04:18.239
l plus or minus must also commute with l
0:04:15.760,0:04:20.320
squared because l plus or minus suggest
0:04:18.239,0:04:21.840
a linear combination of lx and ly
0:04:20.320,0:04:24.800
and each of these individually commutes
0:04:21.840,0:04:27.600
with l squared so we have this
0:04:24.800,0:04:28.960
the commutator of these two is zero and
0:04:27.600,0:04:32.639
that means that
0:04:28.960,0:04:33.680
recalling the eigenvalue equations
0:04:32.639,0:04:36.479
for l squared
0:04:33.680,0:04:37.120
so just just to recap that we can
0:04:36.479,0:04:40.800
pick
0:04:37.120,0:04:42.400
one of the three our x y or z
0:04:40.800,0:04:44.639
and have a well-defined angular momentum
0:04:42.400,0:04:46.479
in that direction and we can have that
0:04:44.639,0:04:48.160
simultaneously with knowledge of the
0:04:46.479,0:04:49.680
eigenvalue of l squared
0:04:48.160,0:04:51.680
so we write the eigenvalue equations as
0:04:49.680,0:04:54.240
follows
0:04:51.680,0:04:55.600
that is we can define some state
0:04:54.240,0:04:58.320
|l,m>
0:04:55.600,0:04:59.840
where l and m are good quantum numbers
0:04:58.320,0:05:01.919
simultaneously
0:04:59.840,0:05:03.280
l z acting on this state gives us
0:05:01.919,0:05:06.240
hbar m l comm
0:05:03.280,0:05:07.520
m just like we saw up here and l squared
0:05:06.240,0:05:09.520
acting on l comma m
0:05:07.520,0:05:11.520
returns an eigenvalue associated with l
0:05:09.520,0:05:13.840
squared which it turns out is
0:05:11.520,0:05:14.880
best written as h bar squared l(l+1)
0:05:13.840,0:05:18.080
where l
0:05:14.880,0:05:18.960
is an integer and to the point of
0:05:18.080,0:05:21.199
noting that
0:05:18.960,0:05:23.280
l plus or minus commute with l squared
0:05:21.199,0:05:26.639
means that when we raise or lower
0:05:23.280,0:05:28.960
m the eigenvalue associated with lz
0:05:26.639,0:05:30.320
the z projection of the angular momentum
0:05:28.960,0:05:33.360
because after all that's what these
0:05:30.320,0:05:34.560
components are this doesn't affect
0:05:33.360,0:05:37.039
the value of
0:05:34.560,0:05:38.000
little l that is the eigenvalue
0:05:37.039,0:05:39.919
associated with
0:05:38.000,0:05:42.320
l squared the total angular momentum
0:05:39.919,0:05:42.320
squared
0:05:42.560,0:05:47.120
so taking this all into account we can
0:05:45.039,0:05:49.120
see on this basis that
0:05:47.120,0:05:50.479
l squared is the square of the length of
0:05:49.120,0:05:51.600
the angular momentum that's certainly
0:05:50.479,0:05:53.360
what it would be classically
0:05:51.600,0:05:54.800
we can think of it like that quantum
0:05:53.360,0:05:55.360
mechanically as well or rather we can
0:05:54.800,0:05:57.840
think of
0:05:55.360,0:05:59.120
the corresponding eigenvalues that
0:05:57.840,0:06:01.520
way
0:05:59.120,0:06:02.240
and so this is conserved when we raise
0:06:01.520,0:06:04.960
or lower
0:06:02.240,0:06:06.400
m the z projection of angular momentum
0:06:04.960,0:06:08.639
and since these
0:06:06.400,0:06:10.800
l plus or minus are forming taking us
0:06:08.639,0:06:12.319
up or down a ladder of states
0:06:10.800,0:06:14.160
we can see that that ladder should have
0:06:12.319,0:06:17.759
both the top and the bottom
0:06:14.160,0:06:20.800
defined by the length the total
0:06:17.759,0:06:22.400
length of the angular momentum vector
0:06:20.800,0:06:24.160
putting this all together we did use the
0:06:22.400,0:06:26.800
following
0:06:24.160,0:06:27.759
so given that l the eigenvalue
0:06:26.800,0:06:29.919
associated with
0:06:27.759,0:06:30.960
the l squared the total angular momentum
0:06:29.919,0:06:32.880
squared
0:06:30.960,0:06:34.240
this is an integer and it's greater than
0:06:32.880,0:06:35.520
or equal to zero
0:06:34.240,0:06:37.520
because you can think of the total
0:06:35.520,0:06:39.919
length of the angular momentum
0:06:37.520,0:06:41.600
operator squared as corresponding to the
0:06:39.919,0:06:42.000
quantum version of the length squared of
0:06:41.600,0:06:44.319
the
0:06:42.000,0:06:46.160
of the angular momentum so clearly it's
0:06:44.319,0:06:49.440
got to be
0:06:46.160,0:06:52.080
zero or greater
0:06:49.440,0:06:54.319
and then m which corresponds to the z
0:06:52.080,0:06:56.880
projection of the angular momentum
0:06:54.319,0:06:58.240
is also an integer and it must fall
0:06:56.880,0:07:01.919
between minus l
0:06:58.240,0:07:02.960
and l so what we've seen is that
0:07:01.919,0:07:04.960
and we can just think about this
0:07:02.960,0:07:06.479
physically because we have
0:07:04.960,0:07:08.400
l squared corresponding to the length
0:07:06.479,0:07:10.160
squared of some vector
0:07:08.400,0:07:12.080
admittedly it's like a quantum operator
0:07:10.160,0:07:13.360
associated with angular momentum but it
0:07:12.080,0:07:15.520
is nevertheless telling us something
0:07:13.360,0:07:16.960
about a some kind of quantum spinning
0:07:15.520,0:07:20.720
particle
0:07:16.960,0:07:22.720
then so the length it has some
0:07:20.720,0:07:25.039
total value and so the z projection
0:07:22.720,0:07:25.840
as in the projection along some
0:07:25.039,0:07:27.360
direction
0:07:25.840,0:07:30.000
can't be any bigger than the length of
0:07:27.360,0:07:33.199
the vector itself okay so that intuition
0:07:30.000,0:07:34.960
still holds in this quantum case and
0:07:33.199,0:07:36.160
so what we see in terms of these raising
0:07:34.960,0:07:38.080
and lowering operators
0:07:36.160,0:07:39.759
is that unlike the harmonic
0:07:38.080,0:07:41.039
oscillator case where we had a ground
0:07:39.759,0:07:42.400
state and then an infinite number of
0:07:41.039,0:07:45.120
rungs above that state
0:07:42.400,0:07:47.440
evenly spaced in energy in this case we
0:07:45.120,0:07:49.360
have the following scenario
0:07:47.440,0:07:50.800
so we start with the case l equals 2
0:07:49.360,0:07:55.440
which consider this case
0:07:50.800,0:07:59.120
so our |l,m> ket has
0:07:55.440,0:07:59.360
l equals 2 then our values
0:07:59.120,0:08:02.879
of
0:07:59.360,0:08:07.039
m can range from -2 up to 2.
0:08:02.879,0:08:09.840
and to go up the rungs of this ladder
0:08:07.039,0:08:11.599
we use the l plus the angular momentum
0:08:09.840,0:08:14.720
raising operator
0:08:11.599,0:08:14.720
we can do that repeatedly
0:08:16.160,0:08:21.199
and to go back down we use l minus
0:08:23.120,0:08:28.479
like so but remember when we studied the
0:08:28.479,0:08:31.520
raising and lowering operators the
0:08:30.080,0:08:33.680
harmonic oscillator
0:08:31.520,0:08:34.560
if we used the lowering operator on the
0:08:33.680,0:08:37.360
ground state
0:08:34.560,0:08:37.360
we got zero
0:08:38.080,0:08:41.839
and it's not the state zero it's the
0:08:39.440,0:08:43.519
number zero we definitely got zero
0:08:41.839,0:08:44.880
now that's true here as well if we
0:08:43.519,0:08:46.800
try to lower off the bottom of the
0:08:44.880,0:08:49.920
ladder we get zero again
0:08:46.800,0:08:50.959
but actually this time if we try to
0:08:49.920,0:08:54.000
raise
0:08:50.959,0:08:57.360
off the top of the ladder we also get
0:08:54.000,0:08:58.080
zero so the ladder has a finite number
0:08:57.360,0:08:59.279
of rungs
0:08:58.080,0:09:01.839
which you can think about physically
0:08:59.279,0:09:02.800
just because you can't get a larger z
0:09:01.839,0:09:05.600
projection
0:09:02.800,0:09:06.240
than the total length of the vector
0:09:06.240,0:09:09.200
and so that's kind of what's happening
0:09:07.440,0:09:10.399
here so we now have a ladder with a
0:09:09.200,0:09:12.160
finite number of rungs
0:09:10.399,0:09:13.839
where the number of rungs is specified
0:09:12.160,0:09:16.800
by l the
0:09:13.839,0:09:17.120
quantum number associated with the
0:09:16.800,0:09:18.880
l
0:09:17.120,0:09:23.839
squared total angular momentum squared
0:09:18.880,0:09:23.839
operator okay thank you for your time
V10.1 Spherically symmetric potentials (angular solution)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
the time independent Schrödinger equation for spherically symmetric potentials. Separating the equation into radial and angular parts, and separating the angular part into polar and azimuthal parts. General form of solutions as spherical harmonics. Continued in video V10.2.
0:00:00.240,0:00:03.679
hello in this video we're going to take
0:00:02.080,0:00:05.839
a look at spherically symmetric
0:00:03.679,0:00:09.120
potentials in three dimensions
0:00:05.839,0:00:11.759
so potentials of the following form
0:00:09.120,0:00:12.880
that is V which is usually a function of
0:00:11.759,0:00:15.200
x y and z
0:00:12.880,0:00:16.960
or r theta and phi is now only a
0:00:15.200,0:00:19.760
function of the radial coordinate
0:00:16.960,0:00:20.320
r so our time independent Schroedinger
0:00:19.760,0:00:21.760
equation
0:00:20.320,0:00:23.279
in general in three dimensions in
0:00:21.760,0:00:25.199
spherically polar coordinates takes the
0:00:23.279,0:00:27.760
following form
0:00:25.199,0:00:29.119
as we saw in a previous video so now our
0:00:27.760,0:00:32.080
potential term is only going to be a
0:00:29.119,0:00:32.080
function of r
0:00:32.320,0:00:37.440
giving us the following so our time
0:00:35.440,0:00:39.680
dependent wave function psi
0:00:37.440,0:00:41.360
we know can always be separated into a
0:00:39.680,0:00:44.320
time dependent part and a spatially
0:00:41.360,0:00:46.480
dependent part
0:00:44.320,0:00:48.239
where the spatially dependent part let's
0:00:46.480,0:00:50.000
write it as
0:00:48.239,0:00:51.440
a slightly fancier phi just to
0:00:50.000,0:00:54.719
distinguish it from
0:00:51.440,0:00:56.399
the polar coordinate phi
0:00:54.719,0:00:58.000
but now in the case of a spherically
0:00:56.399,0:01:00.079
symmetric potential
0:00:58.000,0:01:02.320
when we have this potential as a
0:01:00.079,0:01:05.920
function of r only
0:01:02.320,0:01:07.920
we can additionally separate the
0:01:05.920,0:01:09.840
radial part from the angular parts as
0:01:07.920,0:01:13.360
follows
0:01:09.840,0:01:14.799
defining varphi(x) where x is the
0:01:13.360,0:01:18.640
three dimensional
0:01:14.799,0:01:21.520
vector is R(r) multiplying
0:01:18.640,0:01:22.240
Y(theta,phi) when we substitute this
0:01:21.520,0:01:25.759
form
0:01:22.240,0:01:27.280
back into the time independent Schroedinger
0:01:25.759,0:01:30.560
equation up here
0:01:27.280,0:01:33.119
we get the following where
0:01:30.560,0:01:33.600
the time dependent part t of course we
0:01:33.119,0:01:34.960
can
0:01:33.600,0:01:37.119
neglect in this equation the time
0:01:34.960,0:01:40.560
independent equation
0:01:37.119,0:01:41.119
and r and y have been substituted in in
0:01:40.560,0:01:44.399
place of
0:01:41.119,0:01:48.000
phi but
0:01:44.399,0:01:48.560
here L squared acts only on y because
0:01:48.560,0:01:51.680
the angular momentum squared
0:01:50.079,0:01:53.439
operator only cares about
0:01:51.680,0:01:55.680
theta and phi coordinates doesn't care
0:01:53.439,0:01:57.439
about r
0:01:55.680,0:01:59.119
and so r has pulled through the l
0:01:57.439,0:02:01.119
squared and similarly over here
0:01:59.119,0:02:02.719
the rate these derivatives the radial
0:02:01.119,0:02:06.640
part acts only on r
0:02:02.719,0:02:08.399
not on y so we can divide through by r y
0:02:06.640,0:02:10.720
to give the following and then we can
0:02:08.399,0:02:14.640
multiply through by 2m
0:02:10.720,0:02:16.560
small r squared to give this result
0:02:14.640,0:02:17.920
where the first term here is only a
0:02:16.560,0:02:19.599
function of r
0:02:17.920,0:02:21.760
the second term is only a function of
0:02:19.599,0:02:24.400
theta and phi and third term
0:02:21.760,0:02:25.280
is again only a function of r so we now
0:02:24.400,0:02:28.160
have successfully
0:02:25.280,0:02:29.440
separated our equation again where
0:02:28.160,0:02:31.120
we have the following if we just
0:02:29.440,0:02:34.160
rearrange things slightly and let's get
0:02:31.120,0:02:36.480
these out of the way for a second
0:02:34.160,0:02:37.599
where the term dependent on theta and
0:02:36.480,0:02:39.920
phi is over here
0:02:37.599,0:02:40.640
the terms dependent on r are all over
0:02:39.920,0:02:42.239
here
0:02:40.640,0:02:43.840
and since these must equal each other
0:02:42.239,0:02:45.760
for all c sub phi and
0:02:43.840,0:02:48.640
r they must both be equal to the same
0:02:45.760,0:02:51.040
constant which we can define to be
0:02:48.640,0:02:52.400
h bar squared k squared where k is
0:02:51.040,0:02:54.800
some dimensionless number
0:02:52.400,0:02:56.239
because l has the units of h bar of
0:02:54.800,0:02:58.239
angular momentum
0:02:56.239,0:02:59.280
so we have two separate equations that
0:02:58.239,0:03:02.319
we can work with here
0:02:59.280,0:03:03.280
let's call this one equation one this
0:03:02.319,0:03:08.159
term
0:03:03.280,0:03:08.159
equals this constant and equation two
0:03:09.040,0:03:13.120
this whole term equals the same constant
0:03:12.000,0:03:13.920
so we'll work with these parts
0:03:13.120,0:03:16.319
separately
0:03:13.920,0:03:18.319
so first let's look at the angular part
0:03:16.319,0:03:20.640
equation one
0:03:18.319,0:03:22.000
so our angular equation tells us that
0:03:20.640,0:03:25.280
the angular momentum
0:03:22.000,0:03:26.480
squared operator L squared acting on y
0:03:25.280,0:03:28.720
of theta and phi
0:03:26.480,0:03:30.159
equals [...]
0:03:28.720,0:03:31.760
theta and phi
0:03:30.159,0:03:33.200
we've seen in a previous video or you
0:03:31.760,0:03:33.920
can just work it out there's an angular
0:03:33.200,0:03:35.519
coordinates
0:03:33.920,0:03:37.760
the angular momentum squared operator
0:03:35.519,0:03:40.799
can be written as follows
0:03:37.760,0:03:42.879
just expanding the operator here if
0:03:40.799,0:03:43.440
we multiply through this entire equation
0:03:42.879,0:03:47.840
by sine
0:03:43.440,0:03:47.840
squared theta we get the following
0:03:47.920,0:03:51.440
where i've also divided through by the h
0:03:49.360,0:03:55.040
of r squared on both sides
0:03:51.440,0:03:55.599
so this y is a function of theta and
0:03:55.040,0:03:57.680
phi
0:03:55.599,0:04:01.120
but actually this equation is once again
0:03:57.680,0:04:03.280
separable using the following definition
0:04:01.120,0:04:04.720
so to find y of theta and phi to be a
0:04:03.280,0:04:07.200
function p of theta
0:04:04.720,0:04:10.400
multiplying a function f of phi we
0:04:07.200,0:04:12.000
insert that into the equation to find
0:04:10.400,0:04:15.519
and in the usual way we divide
0:04:12.000,0:04:17.280
through by [...] to get the result
0:04:15.519,0:04:19.280
where i've rearranged slightly to get
0:04:17.280,0:04:20.479
only functions of theta on this side
0:04:19.280,0:04:22.720
and so these derivatives again have
0:04:20.479,0:04:23.759
become total derivatives only a
0:04:22.720,0:04:26.560
function of phi
0:04:23.759,0:04:28.160
on the right hand side so once again
0:04:26.560,0:04:29.360
since these must be equal to each other
0:04:28.160,0:04:30.880
for all theta and phi
0:04:29.360,0:04:34.160
they must both equal to the same
0:04:30.880,0:04:36.000
constant which we can define as follows
0:04:34.160,0:04:37.360
it's equal to m squared where this is
0:04:36.000,0:04:38.960
the conventional choice
0:04:37.360,0:04:40.320
but as we've seen in previous videos m
0:04:38.960,0:04:41.840
is not the mass now it's a little bit
0:04:40.320,0:04:44.240
confusing
0:04:41.840,0:04:45.840
it's just some constant we're using to
0:04:44.240,0:04:47.280
separate these equations
0:04:45.840,0:04:49.520
later on we'll find it turns out to be
0:04:47.280,0:04:50.880
an integer but for now it's just some
0:04:49.520,0:04:52.720
number to be determined
0:04:50.880,0:04:54.639
which is not the mass but unfortunately
0:04:52.720,0:04:57.520
is usually given the same symbol
0:04:54.639,0:04:58.720
so once again we have two equations here
0:04:57.520,0:05:02.720
let's call them
0:04:58.720,0:05:05.520
a and b
0:05:02.720,0:05:07.280
so a is that this part this function of
0:05:05.520,0:05:09.360
phi is equal to m squared
0:05:07.280,0:05:11.280
equation b is that this function of
0:05:09.360,0:05:12.639
theta is equal to m squared
0:05:11.280,0:05:14.800
and we need to deal with them both
0:05:12.639,0:05:16.400
separately so within
0:05:14.800,0:05:19.360
the angular equation we're going to move
0:05:16.400,0:05:22.080
to look at a and b separately
0:05:19.360,0:05:22.400
so first the polar equation a tells us
0:05:22.080,0:05:24.080
that
0:05:22.400,0:05:25.520
the second derivative of f with respect
0:05:24.080,0:05:27.759
to phi squared
0:05:25.520,0:05:29.680
is equal so minus that is equal to m
0:05:27.759,0:05:31.199
squared f of phi
0:05:29.680,0:05:33.680
so we can see that the following
0:05:31.199,0:05:36.400
solution solves this
0:05:33.680,0:05:38.160
so f of phi is technically proportional
0:05:36.400,0:05:41.280
to e to the plus or minus i m
0:05:38.160,0:05:42.479
phi but the constant of proportionality
0:05:41.280,0:05:44.880
won't matter because we
0:05:42.479,0:05:46.560
can define
0:05:44.880,0:05:48.639
our
0:05:46.560,0:05:49.840
normalization as part of the the full
0:05:48.639,0:05:51.680
solution in a second
0:05:49.840,0:05:53.280
so let's just say this is equal for now
0:05:51.680,0:05:55.120
this is really a definitional choice
0:05:53.280,0:05:56.479
we could put a normalization here and it
0:05:55.120,0:05:57.280
would change our normalization on the
0:05:56.479,0:06:00.639
other parts
0:05:57.280,0:06:01.440
so this is a convenient choice and
0:06:00.639,0:06:04.000
this gives us
0:06:01.440,0:06:04.880
a condition on the m the separation
0:06:04.000,0:06:08.000
constant
0:06:04.880,0:06:12.000
because we require that the modulus of f
0:06:08.000,0:06:15.360
of phi squared is an observable quantity
0:06:12.000,0:06:18.880
and so this should be single valued
0:06:15.360,0:06:20.720
so for any value of phi we get a unique
0:06:18.880,0:06:23.120
value of modulus of f of phi
0:06:20.720,0:06:24.960
squared so through a process of
0:06:23.120,0:06:25.919
reasoning this leads us to the result
0:06:24.960,0:06:28.479
that
0:06:25.919,0:06:29.120
when we vary our polar coordinate phi by
0:06:28.479,0:06:31.440
a full
0:06:29.120,0:06:32.400
turn of 2 pi we should get an exactly
0:06:31.440,0:06:34.960
equivalent
0:06:32.400,0:06:36.400
function back in order for this the
0:06:34.960,0:06:37.280
observable quantity to remain single
0:06:36.400,0:06:39.039
valued
0:06:37.280,0:06:40.319
and if this is true this places the
0:06:39.039,0:06:44.479
constraint on m
0:06:40.319,0:06:47.919
that e to the two pi i m must equal one
0:06:44.479,0:06:50.880
and so we have that m must be an integer
0:06:47.919,0:06:53.039
and let's box that
0:06:50.880,0:06:55.199
so it's important to note that if we
0:06:53.039,0:06:55.919
take the z projection of the angular
0:06:55.199,0:06:57.599
momentum
0:06:55.919,0:06:59.440
as follows let's just get rid of this a
0:06:57.599,0:07:01.520
bit for a second
0:06:59.440,0:07:02.720
so the z projection of angular momentum
0:07:01.520,0:07:06.400
acting on the state f
0:07:02.720,0:07:07.759
of phi in spherical polar coordinates is
0:07:06.400,0:07:10.960
minus i h bar d
0:07:07.759,0:07:11.599
by d phi partial holding r and theta and
0:07:10.960,0:07:14.000
time
0:07:11.599,0:07:15.039
constant acting on phi but when we look
0:07:14.000,0:07:18.400
at the form we have here
0:07:15.039,0:07:18.880
we see that this simply returns plus or
0:07:18.400,0:07:21.680
minus
0:07:18.880,0:07:22.960
h bar m acting on f of phi and so you
0:07:21.680,0:07:26.080
can see that f of i
0:07:22.960,0:07:29.360
is actually the eigen function of the lz
0:07:26.080,0:07:31.360
operator and so m here is
0:07:29.360,0:07:32.400
some quantum number associated with
0:07:31.360,0:07:34.160
that operator
0:07:32.400,0:07:36.880
and it's what we call the magnetic
0:07:34.160,0:07:36.880
quantum number
0:07:37.440,0:07:44.160
so it's convenient to label our
0:07:40.720,0:07:48.240
wave function solutions f by the integer
0:07:44.160,0:07:50.479
m and we'll do that from now on
0:07:48.240,0:07:52.319
as m is a good quantum number to
0:07:50.479,0:07:55.520
describe
0:07:52.319,0:07:59.599
our solutions okay let's take a look at
0:07:55.520,0:08:00.319
part b so b is called the azimuthal
0:07:59.599,0:08:01.759
equation
0:08:00.319,0:08:03.840
referring to the azimuthal coordinate
0:08:01.759,0:08:06.240
theta it reads as follows
0:08:03.840,0:08:07.440
just copying from a couple of
0:08:06.240,0:08:09.919
boards ago
0:08:07.440,0:08:12.639
it's an ordinary
0:08:09.919,0:08:14.800
differential equation in terms of theta
0:08:12.639,0:08:16.400
and as usual we have a complicated
0:08:14.800,0:08:18.720
ordinary differential equation the trick
0:08:16.400,0:08:20.879
is just to look up previous equations
0:08:18.720,0:08:23.199
and massage this into the correct
0:08:20.879,0:08:26.319
form and we can do that in this case
0:08:23.199,0:08:29.520
in fact this is a form
0:08:26.319,0:08:31.280
of what's called Legendre's equation
0:08:29.520,0:08:34.800
and the solutions in this particular
0:08:31.280,0:08:34.800
case take the following form
0:08:35.039,0:08:38.880
we write them as P superscript m
0:08:37.360,0:08:41.360
subscript l
0:08:38.880,0:08:43.839
superscript m is the same m that we used
0:08:41.360,0:08:46.240
in the polar equation
0:08:43.839,0:08:47.839
and it's a function of cos theta and
0:08:46.240,0:08:49.839
these things are called the associated
0:08:47.839,0:08:51.839
legendre polynomials and you can
0:08:49.839,0:08:52.880
look those up it's possible to derive
0:08:51.839,0:08:53.680
this solution but really all you're
0:08:52.880,0:08:55.519
doing is
0:08:53.680,0:08:57.760
massaging this into a form of an
0:08:55.519,0:09:02.000
equation that was previously studied
0:08:57.760,0:09:04.640
so the l here is a new quantum number
0:09:02.000,0:09:05.040
it's defined as follows so in terms of
0:09:04.640,0:09:08.320
our
0:09:05.040,0:09:08.720
constant k squared k squared is l l plus
0:09:08.320,0:09:11.680
one
0:09:08.720,0:09:13.680
and again this choice of how to write
0:09:11.680,0:09:16.399
things is just really part of the
0:09:13.680,0:09:18.320
already worked out theory of the
0:09:16.399,0:09:20.160
associated legendre polynomial so this
0:09:18.320,0:09:22.880
is conventional
0:09:20.160,0:09:24.480
and we call this quantum number l either
0:09:22.880,0:09:27.760
the orbital quantum number
0:09:24.480,0:09:31.200
or the azimuthal quantum number
0:09:27.760,0:09:35.440
and l can take the value
0:09:31.200,0:09:37.360
of any integer zero or
0:09:35.440,0:09:38.720
greater. l is an integer greater than
0:09:37.360,0:09:41.360
or equal to zero
0:09:38.720,0:09:42.399
but actually the form of m is now
0:09:41.360,0:09:44.720
further constrained
0:09:42.399,0:09:46.160
before we saw that m must be an integer
0:09:44.720,0:09:47.279
but in fact when we look into the
0:09:46.160,0:09:50.959
theory of associated
0:09:47.279,0:09:54.800
Legendre polynomials we find that the
0:09:50.959,0:09:58.160
m must be between minus l and l
0:09:54.800,0:09:58.480
so let's just box these two so m and l
0:09:58.160,0:10:00.640
here
0:09:58.480,0:10:02.800
are the quantum numbers associated with
0:10:00.640,0:10:04.399
the squared angular momentum operator
0:10:02.800,0:10:05.760
so putting all this back together we can
0:10:04.399,0:10:07.600
get the solutions to the angular
0:10:05.760,0:10:09.920
equation
0:10:07.600,0:10:11.440
so recall that our equation was the
0:10:09.920,0:10:13.839
angular momentum squared
0:10:11.440,0:10:15.680
operator acting on y of theta and phi is
0:10:13.839,0:10:17.920
equal to h bar squared k squared
0:10:15.680,0:10:20.480
y of theta and phi and we now see that
0:10:17.920,0:10:23.760
the solutions are these
0:10:20.480,0:10:26.399
y subscript l superscript m
0:10:23.760,0:10:27.519
theta and phi returning the eigenvalues
0:10:26.399,0:10:30.880
h bar squared l
0:10:27.519,0:10:32.399
l plus one multiplying the same
0:10:30.880,0:10:35.600
and there's a second operator which
0:10:32.399,0:10:37.680
commutes with l squared which is lz
0:10:35.600,0:10:39.120
returning eigenvalues h bar m
0:10:37.680,0:10:41.440
multiplying the same
0:10:39.120,0:10:43.839
of course we could have used x and y
0:10:41.440,0:10:45.680
at lx or ly in place of lz
0:10:43.839,0:10:46.959
but lz takes the simplest form in
0:10:45.680,0:10:48.640
spherical polar coordinates so it's
0:10:46.959,0:10:53.200
convenient to work with
0:10:48.640,0:10:55.839
and in both cases Y_{lm} is defined as
0:10:53.200,0:10:57.360
P_{lm} of cos theta the associated
0:10:55.839,0:10:59.519
Legendre polynomials
0:10:57.360,0:11:00.800
multiplying e to the plus or minus i
0:10:59.519,0:11:02.160
m phi
0:11:00.800,0:11:04.000
multiplying some normalization which
0:11:02.160,0:11:07.200
we'll take a look at in a second
0:11:04.000,0:11:08.959
and the result here y lm
0:11:07.200,0:11:12.000
of theta and phi is what is called the
0:11:08.959,0:11:14.160
spherical harmonics
0:11:12.000,0:11:15.120
and these can be looked up and
0:11:14.160,0:11:19.040
in general
0:11:15.120,0:11:19.680
okay and again this is a generally
0:11:19.680,0:11:22.800
used notation and you can look
0:11:21.200,0:11:23.279
up the spherical harmonics and expect to
0:11:22.800,0:11:24.640
find
0:11:23.279,0:11:26.399
the particular forms of these
0:11:24.640,0:11:28.240
functions
0:11:26.399,0:11:29.600
let's just look at this again
0:11:28.240,0:11:32.720
a little bit in in the ket
0:11:29.600,0:11:33.600
formalism and direct notation so it's
0:11:32.720,0:11:37.040
convenient to
0:11:33.600,0:11:38.320
define our spherical harmonics ylm of
0:11:37.040,0:11:40.880
theta and phi
0:11:38.320,0:11:43.279
as the projection into the theta phi
0:11:40.880,0:11:46.480
position basis
0:11:43.279,0:11:48.240
of the ket l comma m because l and m
0:11:46.480,0:11:49.839
are good quantum numbers so we can label
0:11:48.240,0:11:51.120
a ket with them at the same time and
0:11:49.839,0:11:54.800
that makes sense
0:11:51.120,0:11:54.800
so our equations then take the form
0:11:55.120,0:11:58.560
that we've just seen on the previous
0:11:56.240,0:12:01.760
board now in direct notation
0:11:58.560,0:12:04.560
so we have that these
0:12:01.760,0:12:04.959
eigen states of l squared and L_z
0:12:04.560,0:12:08.639
form
0:12:04.959,0:12:08.639
a complete orthonormal basis
0:12:08.880,0:12:14.560
so they are also normal the
0:12:12.000,0:12:17.360
inner product of l prime m prime with
0:12:14.560,0:12:18.880
l m it gives a product of Kronecker
0:12:17.360,0:12:21.839
deltas as usual
0:12:18.880,0:12:22.959
and as usual we can expand this into the
0:12:21.839,0:12:26.160
position basis
0:12:22.959,0:12:28.720
into spherical polar coordinates
0:12:26.160,0:12:31.040
using an insertion of a complete set of
0:12:28.720,0:12:34.399
states
0:12:31.040,0:12:35.839
that is just inserting theta comma phi
0:12:35.839,0:12:40.880
this ket which hopefully makes sense
0:12:38.959,0:12:42.320
this is up here that projects us into
0:12:40.880,0:12:44.480
the theta phi space
0:12:42.320,0:12:45.920
and integrating over theta and phi
0:12:44.480,0:12:47.600
but we need to use the correct volume
0:12:45.920,0:12:49.600
element here the Jacobian that takes us
0:12:47.600,0:12:52.560
from cartesians to spherical polars
0:12:49.600,0:12:55.120
so we have a sine theta term here and so
0:12:52.560,0:12:58.399
these take the form
0:12:55.120,0:13:00.399
y l prime m prime of theta and phi star
0:12:58.399,0:13:03.040
because this is now a complex function
0:13:00.399,0:13:04.720
multiplying ylm theta phi integrated
0:13:03.040,0:13:07.440
over the volume element and we get these
0:13:04.720,0:13:09.279
two Kronecker deltas for ll prime and mm
0:13:07.440,0:13:11.680
prime
0:13:09.279,0:13:12.560
so we've shown that ylms the spherical
0:13:11.680,0:13:15.279
harmonics
0:13:12.560,0:13:16.560
are or form an orthonormal basis and in
0:13:15.279,0:13:17.839
fact additionally they form a complete
0:13:16.560,0:13:19.200
orthonormal basis
0:13:17.839,0:13:21.839
meaning we can write any function of
0:13:19.200,0:13:21.839
this form
0:13:21.920,0:13:26.720
that is f of theta and phi can be
0:13:24.079,0:13:29.279
written as a sum from m is minus l to l
0:13:26.720,0:13:30.480
l from zero to infinity of y ln c
0:13:29.279,0:13:32.399
squared phi
0:13:30.480,0:13:34.800
multiplied by some complex coefficients
0:13:32.399,0:13:35.519
flm to be determined in the usual manner
0:13:34.800,0:13:38.720
by
0:13:35.519,0:13:40.480
using the orthonormality of the ylms
0:13:38.720,0:13:42.399
so that's a convenient way to write
0:13:40.480,0:13:44.160
any function which does not depend on
0:13:42.399,0:13:44.639
the radial coordinates only on theta and
0:13:44.160,0:13:46.839
phi
0:13:44.639,0:13:48.079
so things that live on the surfaces of
0:13:46.839,0:13:50.079
spheres
0:13:48.079,0:13:51.680
so that's the angular equation solved in
0:13:50.079,0:13:53.120
the next video we'll take a look at
0:13:51.680,0:13:56.079
solving the radial equation
0:13:53.120,0:13:56.079
thanks for your time
V10.2 Spherically symmetric potentials (radial solution)
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
(continuing from V10.1) the radial part of the TISE in spherically symmetric potentials. Rewriting in terms of the one-dimensional TISE with an effective potential with centrifugal barrier term; general form of solutions to the TISE in spherically symmetric potentials.
0:00:00.320,0:00:03.120
hello in this video following on from
0:00:02.639,0:00:04.960
the last
0:00:03.120,0:00:06.319
we're going to take a look at solving
0:00:04.960,0:00:08.320
the radial equation
0:00:06.319,0:00:10.960
as part of solving spherically symmetric
0:00:08.320,0:00:13.120
potentials
0:00:10.960,0:00:14.000
so the radial equation takes this form
0:00:14.000,0:00:17.279
we have the separation constant k
0:00:16.080,0:00:19.119
squared over here
0:00:17.279,0:00:20.320
but we saw when solving the angular
0:00:19.119,0:00:22.160
part that
0:00:20.320,0:00:23.600
it's also natural to write k squared
0:00:22.160,0:00:25.439
as l(l+1)
0:00:23.600,0:00:26.720
just for historical reasons to do
0:00:25.439,0:00:28.560
with the particular form of the
0:00:26.720,0:00:31.519
solutions being the associated
0:00:28.560,0:00:32.800
Legendre polynomials so we can
0:00:31.519,0:00:33.920
equally well write this in the following
0:00:32.800,0:00:35.200
way
0:00:33.920,0:00:36.719
and it just depends on which branch of
0:00:35.200,0:00:37.200
physics you come from as to whether you
0:00:36.719,0:00:40.079
use
0:00:37.200,0:00:41.280
k squared here or l(l+1) so they're
0:00:40.079,0:00:43.280
just different choices
0:00:41.280,0:00:44.800
of quantum number that you can use in
0:00:43.280,0:00:48.160
this we've gone for l
0:00:44.800,0:00:51.039
so we've labeled the functions R(r)
0:00:48.160,0:00:52.640
by the quantum number l here a
0:00:51.039,0:00:56.879
particularly convenient choice
0:00:52.640,0:00:58.800
of a solution to R(r) is as follows
0:00:56.879,0:00:59.920
if we define some new function
0:00:58.800,0:01:03.120
chi(r)/r
0:00:59.920,0:01:05.360
to be equal to R(r) we can rewrite the
0:01:03.120,0:01:07.360
equation in the following form
0:01:05.360,0:01:09.200
so you can write it like this where
0:01:07.360,0:01:11.680
V_effective(r)
0:01:09.200,0:01:12.799
is a potential defined as follows the
0:01:11.680,0:01:15.280
original potential
0:01:12.799,0:01:16.400
plus h bar squared l(l+1)/2
0:01:15.280,0:01:18.320
m r squared
0:01:16.400,0:01:21.439
and this term here is what's called
0:01:18.320,0:01:23.280
the centrifugal barrier term
0:01:21.439,0:01:25.280
the convenience of writing this
0:01:23.280,0:01:26.240
redefinition is that this equation up
0:01:25.280,0:01:27.439
here
0:01:26.240,0:01:28.720
is nothing other than our
0:01:27.439,0:01:29.920
one-dimensional time-independent
0:01:28.720,0:01:31.600
schroedinger equation
0:01:29.920,0:01:34.159
it's just that we've had to redefine our
0:01:31.600,0:01:35.600
potential to V_effective
0:01:34.159,0:01:39.200
so putting everything back together we
0:01:35.600,0:01:41.040
get the ultimate solution to the problem
0:01:39.200,0:01:42.399
the radially symmetric hamiltonian
0:01:41.040,0:01:44.560
acting on phi l m
0:01:42.399,0:01:45.680
of x the three dimensional vector is
0:01:44.560,0:01:48.240
given by this
0:01:45.680,0:01:50.079
where the labels phi l m have allowed us
0:01:48.240,0:01:50.960
to rewrite the angular momentum squared
0:01:50.079,0:01:53.520
operator
0:01:50.960,0:01:54.079
as its eigenvalue h bar squared
0:01:53.520,0:01:56.960
l(l+1)
0:01:54.079,0:01:57.439
acting on the eigenfunction and the
0:01:56.960,0:02:01.200
states
0:01:57.439,0:02:04.399
phi l m are defined as
0:02:01.200,0:02:05.680
R labeled by the quantum number l
0:02:04.399,0:02:07.920
as a function of radius
0:02:05.680,0:02:09.920
multiplying spherical harmonics y l m as
0:02:07.920,0:02:11.520
a function of theta and phi
0:02:09.920,0:02:13.200
in terms of normalization we'd like the
0:02:11.520,0:02:14.480
following condition
0:02:13.200,0:02:16.640
the integral across all of three
0:02:14.480,0:02:19.680
dimensional space of phi l m
0:02:16.640,0:02:21.120
of x modulus squared should equal one
0:02:19.680,0:02:22.560
we get a choice as to how we distribute
0:02:21.120,0:02:23.360
the normalization amongst the different
0:02:22.560,0:02:25.040
parts
0:02:23.360,0:02:27.440
but a convenient choice to make is
0:02:25.040,0:02:28.560
this the integral of the radial part
0:02:27.440,0:02:31.680
modulus squared
0:02:28.560,0:02:34.000
across zero to infinity of r
0:02:31.680,0:02:35.840
is one where we've used the jacobian of
0:02:34.000,0:02:37.360
r squared which just takes us from
0:02:35.840,0:02:38.560
cartesian coordinates to spherical
0:02:37.360,0:02:41.760
polars
0:02:38.560,0:02:44.800
and for the angular part
0:02:41.760,0:02:45.599
this then must also equal one but we
0:02:44.800,0:02:47.840
notice that
0:02:45.599,0:02:49.760
this first term the phi dependent term
0:02:47.840,0:02:52.239
the modulus of e to the plus or minus i
0:02:49.760,0:02:54.000
m phi where m is an integer
0:02:52.239,0:02:57.120
modulus square is always
0:02:54.000,0:02:58.959
one anyway so we just get the result
0:02:57.120,0:03:00.319
as follows so this is a particular
0:02:58.959,0:03:02.000
choice of normalization we can make
0:03:00.319,0:03:03.280
which is particularly convenient
0:03:02.000,0:03:04.800
so in the next video we'll take a look
0:03:03.280,0:03:07.040
at a particular instance of a
0:03:04.800,0:03:10.440
spherically symmetric potential
0:03:07.040,0:03:13.440
the hydrogen atom thank you for your
0:03:10.440,0:03:13.440
time
V10.3 The hydrogen atom
This channel, Introductory Quantum Mechanics, is a set of videos aimed at second-year physics undergraduates.
This video:
Solution to the TDSE for the electron in a hydrogen atom. Separating the equation into radial and angular parts; converting the radial equation into Laguerre's equation, and solving with generalised Laguerre polynomials; the wavefunction of the electron in the hydrogen atom, including quantum numbers.
0:00:00.160,0:00:03.439
hello in this video we're going to take
0:00:01.920,0:00:04.960
a look at the hydrogen atom
0:00:03.439,0:00:07.279
one of the most important applications
0:00:04.960,0:00:09.200
of quantum mechanics because it provided
0:00:07.279,0:00:11.040
a full solution to one of the original
0:00:09.200,0:00:12.000
experimental motivations of developing
0:00:11.040,0:00:13.360
the subject
0:00:12.000,0:00:16.480
so the time-independent Schroedinger
0:00:13.360,0:00:17.840
equation is as follows so we have a
0:00:16.480,0:00:19.840
kinetic energy term
0:00:17.840,0:00:21.920
this rather than the electron mass here
0:00:19.840,0:00:24.080
this is the reduced mass of the electron
0:00:21.920,0:00:26.400
nucleus system
0:00:24.080,0:00:27.199
where m_n would usually be the mass of
0:00:26.400,0:00:29.199
the proton
0:00:27.199,0:00:31.039
but we could have deuterium where we
0:00:29.199,0:00:31.760
have a proton and a neutron in the
0:00:31.039,0:00:33.200
nucleus
0:00:31.760,0:00:35.120
or tritium where we have a proton and
0:00:33.200,0:00:36.800
two neutrons or we could even consider
0:00:35.120,0:00:39.440
things such as positronium where you
0:00:36.800,0:00:41.920
have an electron positron pair
0:00:39.440,0:00:42.719
in a bound state so that's the kinetic
0:00:41.920,0:00:46.800
energy term
0:00:42.719,0:00:49.200
the potential energy term is due to the
0:00:46.800,0:00:50.079
electrostatic attraction between the
0:00:49.200,0:00:53.199
electron
0:00:50.079,0:00:54.160
and the nucleus so it's describing the
0:00:53.199,0:00:57.600
behavior of
0:00:54.160,0:01:01.039
the electron in the hydrogen atom
0:00:57.600,0:01:02.640
we can see that we have bound states we
0:01:01.039,0:01:03.280
have negative potential for all values
0:01:02.640,0:01:05.199
of r
0:01:03.280,0:01:06.960
so we have an infinite set of bound
0:01:05.199,0:01:10.240
states in this system
0:01:06.960,0:01:11.520
and we have a spherically symmetric
0:01:10.240,0:01:13.760
potential so let's write those two
0:01:11.520,0:01:14.799
things down
0:01:13.760,0:01:16.960
the fact that we have a spherically
0:01:14.799,0:01:19.119
symmetric potential means that we can
0:01:16.960,0:01:20.479
bring to bear all the tools we developed
0:01:19.119,0:01:22.479
in the previous video where we studied
0:01:20.479,0:01:24.159
such cases in generality so let's take a
0:01:22.479,0:01:25.920
look at the solutions here
0:01:24.159,0:01:27.920
so we can substitute an ansatz of the
0:01:25.920,0:01:29.759
form
0:01:27.920,0:01:32.000
chi is a function of r divided by the
0:01:29.759,0:01:34.320
radius r multiplying the spherical
0:01:32.000,0:01:35.040
harmonics ylm as a function of theta and
0:01:34.320,0:01:36.960
phi
0:01:35.040,0:01:38.960
substituting it in we get the following
0:01:36.960,0:01:41.200
form
0:01:38.960,0:01:42.560
we can simplify a little bit by defining
0:01:41.200,0:01:44.399
some new constants
0:01:42.560,0:01:47.680
so we can define the Bohr radius or
0:01:44.399,0:01:49.840
rather the reduced Bohr radius
0:01:47.680,0:01:51.439
if this mu were the mass of the
0:01:49.840,0:01:54.000
electron this would be the Bohr radius
0:01:51.439,0:01:54.960
its length scale appropriate for the
0:01:54.000,0:01:56.320
system
0:01:54.960,0:01:59.520
and using this we can define a
0:01:56.320,0:02:02.320
dimensionless radius
0:01:59.520,0:02:05.119
r/a_0 and a
0:02:02.320,0:02:07.759
dimensionless energy scale
0:02:05.119,0:02:08.720
given here if we substitute these back
0:02:07.759,0:02:11.920
into the equation
0:02:08.720,0:02:13.840
we get the following where
0:02:11.920,0:02:15.920
i've slightly redefined chi so that it's
0:02:13.840,0:02:19.040
now just a function of rho
0:02:15.920,0:02:20.239
so remember the way to
0:02:19.040,0:02:22.000
try and solve such
0:02:20.239,0:02:23.599
ordinary differential equations is to
0:02:22.000,0:02:25.920
massage them into a form
0:02:23.599,0:02:26.879
where they've already been solved in
0:02:25.920,0:02:30.160
this case
0:02:26.879,0:02:31.280
we can do this by looking at different
0:02:30.160,0:02:33.280
limits
0:02:31.280,0:02:35.519
so we can look at the case that r is
0:02:33.280,0:02:37.040
very large rho is very large
0:02:35.519,0:02:38.640
that is rho is much greater than one
0:02:37.040,0:02:40.959
because it's dimensionless
0:02:38.640,0:02:42.239
in that case we lose this term and this
0:02:40.959,0:02:44.160
term because they're divided by
0:02:42.239,0:02:46.080
rho and rho squared respectively our
0:02:44.160,0:02:48.160
equation becomes
0:02:46.080,0:02:50.000
d squared chi by d rho squared is equal
0:02:48.160,0:02:53.120
to lambda squared chi
0:02:50.000,0:02:55.440
and so the relevant solutions are
0:02:53.120,0:02:56.239
chi goes as e to the minus lambda rho
0:02:55.440,0:02:58.480
for rho
0:02:56.239,0:02:59.599
much greater than one the the other
0:02:58.480,0:03:02.080
solution that we could have had
0:02:59.599,0:03:02.720
would not be normalizable and the
0:03:02.080,0:03:06.080
other limit
0:03:02.720,0:03:06.080
rho is much smaller than one
0:03:06.400,0:03:10.400
this term becomes significantly more
0:03:08.480,0:03:13.360
important than this one or this one
0:03:10.400,0:03:14.400
and so our equation reduces to the
0:03:13.360,0:03:16.720
following form
0:03:14.400,0:03:17.440
and you can check by substituting that
0:03:17.440,0:03:22.879
this form will solve it chi
0:03:20.959,0:03:25.360
as a polynomial goes as rho^(l+1)
0:03:22.879,0:03:28.239
substituting it into here
0:03:25.360,0:03:28.720
you bring down l plus one and l and
0:03:28.239,0:03:30.959
you get
0:03:28.720,0:03:32.159
chi over rho squared and that cancels
0:03:30.959,0:03:35.599
with this term
0:03:32.159,0:03:37.680
so using these two limits we can
0:03:35.599,0:03:38.239
motivate another ansatz to transform our
0:03:37.680,0:03:40.959
equation
0:03:38.239,0:03:42.000
to the following form so we bring the
0:03:40.959,0:03:44.560
two different
0:03:42.000,0:03:46.239
limiting forms out and multiply by
0:03:44.560,0:03:49.280
some unknown function
0:03:46.239,0:03:51.519
and scale it by 2 lambda rho and in fact
0:03:49.280,0:03:55.680
it's convenient to define a new variable
0:03:51.519,0:03:58.959
let's define the variable y as follows
0:03:55.680,0:04:00.879
so it's 2 lambda rho is equal to y
0:03:58.959,0:04:02.159
another form of length scale when we
0:04:00.879,0:04:05.439
substitute
0:04:02.159,0:04:06.000
this form back into our equation up at
0:04:05.439,0:04:09.840
the top
0:04:06.000,0:04:11.920
we transform it into the following
0:04:09.840,0:04:13.439
so it may not look particularly nice but
0:04:11.920,0:04:15.040
it is an equation that's already been
0:04:13.439,0:04:17.440
solved we knew the solutions
0:04:15.040,0:04:19.199
this is in fact what's called Laguerre's
0:04:17.440,0:04:20.959
equation
0:04:19.199,0:04:25.199
and in this case the solutions are what
0:04:20.959,0:04:28.080
are called generalized Laguerre polynomials
0:04:25.199,0:04:28.639
which have written as capital L they
0:04:28.080,0:04:31.040
have two
0:04:28.639,0:04:33.040
indices the lower one is
0:04:31.040,0:04:33.919
n-l-1 in this case the top one is
0:04:33.040,0:04:36.720
2l+1
0:04:33.919,0:04:37.520
and in this case a function of y so the
0:04:37.520,0:04:43.840
generalized Laguerre polynomials you can
0:04:40.160,0:04:43.840
find them from the following formula
0:04:44.080,0:04:48.240
so you can substitute in the
0:04:46.880,0:04:50.560
relevant parts
0:04:48.240,0:04:52.320
and these generalized negative
0:04:50.560,0:04:54.639
polynomials are orthogonal to one
0:04:52.320,0:04:54.639
another
0:04:54.800,0:04:59.199
where orthogonality for these functions
0:04:56.800,0:05:01.840
is defined as an inner product here
0:04:59.199,0:05:03.039
where the weight function defined to be
0:05:01.840,0:05:06.080
y to the alpha
0:05:03.039,0:05:07.600
e to the minus y and in this case
0:05:06.080,0:05:10.639
these ones aren't normalized they have
0:05:07.600,0:05:13.840
this pre-factor
0:05:10.639,0:05:16.400
so they're labeled by
0:05:13.840,0:05:17.199
two labels and those labels are
0:05:16.400,0:05:19.919
two
0:05:17.199,0:05:22.240
quantum numbers l we've seen before
0:05:19.919,0:05:24.720
it's the azimuthal quantum number
0:05:22.240,0:05:26.160
and we're seeing for the first time and
0:05:24.720,0:05:27.440
it's what's called the principal quantum
0:05:26.160,0:05:29.360
number
0:05:27.440,0:05:31.280
so n is an integer and it's greater than
0:05:29.360,0:05:32.160
zero it's called the principal quantum
0:05:31.280,0:05:33.840
number
0:05:32.160,0:05:35.759
and the form of these solutions the
0:05:33.840,0:05:38.000
generalized Laguerre polynomials
0:05:35.759,0:05:39.520
actually places a limit on l which we
0:05:38.000,0:05:43.039
didn't have before
0:05:39.520,0:05:43.759
of the following form so l again an
0:05:43.039,0:05:45.520
integer
0:05:43.759,0:05:46.800
but it must be greater than or equal to
0:05:45.520,0:05:50.400
zero but it must be less than
0:05:46.800,0:05:53.440
n so there's an additional constraint
0:05:50.400,0:05:55.680
coming from the form of these solutions
0:05:53.440,0:05:57.919
if we take a look at the original
0:05:55.680,0:06:01.280
equation of the full solution
0:05:57.919,0:06:02.240
we get this so original time independent
0:06:01.280,0:06:05.360
Schroedinger equation
0:06:02.240,0:06:06.000
here written in Dirac notation the
0:06:05.360,0:06:08.880
state
0:06:06.000,0:06:11.280
sorry the eigen energy E_n is given
0:06:08.880,0:06:14.240
by this
0:06:11.280,0:06:15.600
these form of one over n squared with
0:06:14.240,0:06:17.840
some prefactors
0:06:15.600,0:06:19.360
actually matches the Bohr model of
0:06:17.840,0:06:20.000
the atom we saw at the very start of the
0:06:19.360,0:06:21.680
course
0:06:20.000,0:06:23.840
so that was a kind of phenomenological
0:06:21.680,0:06:24.400
guess this is now the full solution to
0:06:23.840,0:06:26.400
the
0:06:24.400,0:06:28.960
time-independent schrodinger equation
0:06:26.400,0:06:31.840
and it matches Bohr's guess
0:06:28.960,0:06:33.199
and the eigenstate |n,l,m> take the
0:06:31.840,0:06:34.000
following form projected into the
0:06:33.199,0:06:36.000
position basis
0:06:34.000,0:06:38.720
and the cartesian sorry the spherical
0:06:36.000,0:06:41.520
polar coordinates
0:06:38.720,0:06:42.000
which is admittedly rather complicated
0:06:41.520,0:06:44.400
but
0:06:42.000,0:06:46.720
you can substitute in values of n l and
0:06:44.400,0:06:48.000
m and find the corresponding
0:06:46.720,0:06:50.080
wave function which solves the time
0:06:48.000,0:06:53.599
independent schrodinger equation
0:06:50.080,0:06:57.360
so we have three quantum numbers
0:06:53.599,0:06:59.120
hidden in here n l and m are all
0:06:57.360,0:07:01.039
integers
0:06:59.120,0:07:02.400
n which is greater than zero is the
0:07:01.039,0:07:04.319
principal quantum number
0:07:02.400,0:07:05.440
but more specifically it's the shell of
0:07:04.319,0:07:08.080
the electron
0:07:05.440,0:07:09.120
so we can be in shell one two three and
0:07:08.080,0:07:13.440
so on
0:07:09.120,0:07:16.319
l the azimuthal quantum number
0:07:13.440,0:07:16.960
is really the orbital of the electron
0:07:16.319,0:07:18.639
so
0:07:16.960,0:07:20.720
they have special names for the
0:07:18.639,0:07:23.440
different values of l
0:07:20.720,0:07:24.319
so we call l equals zero one two three
0:07:23.440,0:07:27.440
etc
0:07:24.319,0:07:29.120
s p d f g
0:07:27.440,0:07:30.720
and then I think it just continues
0:07:29.120,0:07:32.639
h i j et cetera
0:07:30.720,0:07:34.000
so these are old-fashioned names that
0:07:32.639,0:07:35.440
don't really mean much anymore I think
0:07:34.000,0:07:38.560
it's
0:07:35.440,0:07:40.880
sharp principle diffuse fine g-h-i that
0:07:38.560,0:07:44.080
they they are meaningless after that
0:07:40.880,0:07:47.120
and so for example in the
0:07:44.080,0:07:49.199
n equals one shell the lowest shell
0:07:47.120,0:07:51.360
we can only have the l equals zero
0:07:49.199,0:07:53.039
case we can only have the s wave case
0:07:51.360,0:07:55.440
or the s orbital
0:07:53.039,0:07:56.879
in n equals two we could have l equals
0:07:55.440,0:07:58.240
zero so we can have an s value or we can
0:07:56.879,0:08:00.000
have a p
0:07:58.240,0:08:01.360
sorry we have an s orbital we have a p
0:08:00.000,0:08:02.879
orbital
0:08:01.360,0:08:07.039
and this is how we build up the periodic
0:08:02.879,0:08:09.280
table and finally we have m
0:08:07.039,0:08:10.960
which ranges from minus l to l and is
0:08:09.280,0:08:12.319
called the magnetic quantum number for
0:08:10.960,0:08:13.440
reasons that we won't really go into in
0:08:12.319,0:08:16.319
this course
0:08:13.440,0:08:18.319
but it's the z projection of the
0:08:16.319,0:08:21.599
angular momentum
0:08:18.319,0:08:22.400
okay so if we take the n equals one
0:08:21.599,0:08:25.280
shell
0:08:22.400,0:08:26.400
l must equal m must equal zero and the
0:08:25.280,0:08:28.400
wave function
0:08:26.400,0:08:30.479
is proportional to with some real
0:08:28.400,0:08:33.760
constants of proportionality
0:08:30.479,0:08:34.399
e to the minus r/a_0 so if we
0:08:33.760,0:08:37.279
plot
0:08:34.399,0:08:39.599
the probability density of this function
0:08:37.279,0:08:42.240
as a function of r
0:08:39.599,0:08:43.839
we just have an exponentially decaying
0:08:42.240,0:08:45.920
probability density
0:08:43.839,0:08:47.839
and if we'd plot a surface of
0:08:45.920,0:08:50.240
constant probability density
0:08:47.839,0:08:52.320
it would look like this so it would just
0:08:50.240,0:08:54.800
be a spherical shell
0:08:52.320,0:08:55.680
if we were to request that the modulus
0:08:54.800,0:08:57.600
square of this
0:08:55.680,0:08:58.959
with its pre-factors is equal to some
0:08:57.600,0:09:00.959
specified constant
0:08:58.959,0:09:02.720
then it'll be a sphere for whatever
0:09:00.959,0:09:06.480
value of c we choose
0:09:02.720,0:09:08.800
if we look at the n equals 2 case
0:09:06.480,0:09:10.480
let's take the l equals one
0:09:08.800,0:09:11.600
case and we'll choose m equals zero to
0:09:10.480,0:09:15.040
start with
0:09:11.600,0:09:16.720
it looks like this again omitting the
0:09:15.040,0:09:18.240
constant of proportionality
0:09:16.720,0:09:20.320
if we plot the modulus square as a
0:09:18.240,0:09:21.920
function of radius
0:09:20.320,0:09:24.080
we see that the modulus square of this
0:09:21.920,0:09:24.959
must increase as r squared towards r
0:09:24.080,0:09:26.399
equals zero
0:09:24.959,0:09:28.000
and then it must decay exponentially
0:09:26.399,0:09:29.360
again out towards r
0:09:28.000,0:09:31.600
tending to large values so we have a
0:09:29.360,0:09:34.320
hump and then decrease
0:09:31.600,0:09:35.760
and so if we were to plot a surface of
0:09:34.320,0:09:38.160
constant probability density it would
0:09:35.760,0:09:41.040
look like this
0:09:38.160,0:09:41.440
so when cos squared theta between
0:09:41.040,0:09:43.839
these
0:09:41.440,0:09:45.040
modulus square this because when theta
0:09:43.839,0:09:46.800
is zero that is
0:09:45.040,0:09:48.560
up at the top or when theta is pi at the
0:09:46.800,0:09:52.640
bottom this is a maximum
0:09:48.560,0:09:54.640
in theta and we see that it must be
0:09:52.640,0:09:55.920
zero towards the bottom here and so it
0:09:54.640,0:09:59.519
must be something like this
0:09:55.920,0:10:01.360
so two bulbs along the z axis
0:09:59.519,0:10:05.040
and if we look at the case n equals two
0:10:01.360,0:10:07.600
l equals one m equals one
0:10:05.040,0:10:08.880
the radial dependence is the same as the
0:10:07.600,0:10:12.160
previous case
0:10:08.880,0:10:14.480
so it looks like this again but
0:10:12.160,0:10:15.760
we now have a different angular
0:10:14.480,0:10:17.680
dependence
0:10:15.760,0:10:19.680
and we'll take the modulus square the
0:10:17.680,0:10:22.160
phi term doesn't contribute anything it
0:10:19.680,0:10:25.360
only gives a complex phase
0:10:22.160,0:10:26.880
but in terms of theta we find the result
0:10:25.360,0:10:29.279
which is my attempt to drawing something
0:10:26.880,0:10:31.279
a little bit like a doughnut
0:10:29.279,0:10:32.800
it it doesn't vanish exactly in the
0:10:31.279,0:10:35.040
middle it vanishes at
0:10:32.800,0:10:35.920
zero when r goes to zero this goes to
0:10:35.040,0:10:38.880
zero
0:10:35.920,0:10:39.600
it increases as a quadratic away from
0:10:38.880,0:10:42.720
zero
0:10:39.600,0:10:44.000
but now when theta equals zero up here
0:10:42.720,0:10:47.440
or theta equals pi
0:10:44.000,0:10:49.760
it's zero reaching a maximum down in
0:10:47.440,0:10:51.760
the z equals zero plane
0:10:49.760,0:10:53.600
so it's a bit like a doughnut it sort
0:10:51.760,0:10:55.040
of spreads out to a thick doughnut but
0:10:53.600,0:10:56.560
the doughnut's got a
0:10:55.040,0:10:58.800
very tiny hole right in the
0:10:56.560,0:10:59.519
center okay so those are some of the
0:10:58.800,0:11:01.600
solutions
0:10:59.519,0:11:07.120
and you can work out others yourself
0:11:01.600,0:11:07.120
thank you very much for listening
__