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 0:00:00.480,0:00:05.279 Hello and welcome to this introductory 0:00:02.879,0:00:07.200 video for introductory quantum mechanics 0:00:05.279,0:00:09.360 the set of videos is going to cover a 0:00:07.200,0:00:11.200 second year course in quantum mechanics. 0:00:09.360,0:00:13.440 I'll assume knowledge of a standard 0:00:11.200,0:00:16.000 first year undergraduate physics course 0:00:13.440,0:00:17.600 so subjects such as complex numbers 0:00:16.000,0:00:20.160 differential equations 0:00:17.600,0:00:21.199 vectors and matrices classical mechanics 0:00:20.160,0:00:24.160 and so on 0:00:21.199,0:00:25.359 I'll provide problem sets and notes to 0:00:24.160,0:00:27.519 accompany the videos 0:00:25.359,0:00:30.400 on my website the link to which can be 0:00:27.519,0:00:32.239 found in the youtube channel description 0:00:30.400,0:00:33.920 I'll try to mix things up a little bit 0:00:32.239,0:00:37.120 sometimes I'll be walking along like 0:00:33.920,0:00:37.120 this with my dog Geoffrey 0:00:38.800,0:00:43.040 sometimes I will instead 0:00:43.120,0:00:51.120 be writing at this board sometimes 0:00:46.719,0:00:53.840 i will be recording worked examples 0:00:51.120,0:00:55.680 in a little bit more detail using pen 0:00:53.840,0:00:59.840 and paper 0:00:55.680,0:00:59.840 let's switch back to the field 0:01:03.359,0:01:06.560 with regard to youtube I don't know 0:01:05.360,0:01:07.760 whether you're seeing adverts at the 0:01:06.560,0:01:09.360 start of this video 0:01:07.760,0:01:11.200 if you don't wish to see the adverts you 0:01:09.360,0:01:12.080 can use something such as adblock plus 0:01:11.200,0:01:14.000 which is available 0:01:12.080,0:01:16.080 for all browsers which will stop the 0:01:14.000,0:01:19.280 adverts on youtube 0:01:16.080,0:01:20.240 there's a setting in the bottom right 0:01:19.280,0:01:21.600 hand corner 0:01:20.240,0:01:23.439 which will allow you to change the 0:01:21.600,0:01:24.880 quality of the video the videos are all 0:01:23.439,0:01:26.960 filmed in 1080p 0:01:24.880,0:01:28.240 fully high definition so if at any point 0:01:26.960,0:01:29.680 you can't see what's being written on 0:01:28.240,0:01:31.840 the board for example 0:01:29.680,0:01:32.880 try increasing the quality setting and 0:01:31.840,0:01:34.960 hopefully that should make things 0:01:32.880,0:01:37.040 clearer 0:01:34.960,0:01:38.320 and finally also in the settings 0:01:37.040,0:01:39.840 you'll find that you can increase the 0:01:38.320,0:01:41.360 speed of the videos up to 0:01:39.840,0:01:43.680 two times so if you want to watch things 0:01:41.360,0:01:45.759 a bit faster that should be possible 0:01:43.680,0:01:47.040 so thank you very much for your time and 0:01:45.759,0:01:49.920 I look forward to seeing you in the 0:01:47.040,0:01:49.920 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. 0:00:01.280,0:00:04.000 hello in this video i'm going to give a 0:00:03.520,0:00:05.680 brief 0:00:04.000,0:00:07.040 overview of the history of the subject 0:00:05.680,0:00:08.720 of quantum mechanics 0:00:07.040,0:00:10.240 so quantum mechanics is somewhat unique 0:00:08.720,0:00:12.480 amongst physics courses 0:00:10.240,0:00:13.360 in seemingly requiring a historical 0:00:12.480,0:00:15.120 background 0:00:13.360,0:00:16.880 and i should add as a disclaimer at the 0:00:15.120,0:00:17.760 start that i'm not a historian i'm a 0:00:16.880,0:00:19.279 physicist 0:00:17.760,0:00:21.840 and so you may want to go and check 0:00:19.279,0:00:23.600 these facts yourself 0:00:21.840,0:00:25.519 nevertheless the purpose of the video 0:00:23.600,0:00:27.760 and the take home message from it 0:00:25.519,0:00:28.720 is that however weird things get later 0:00:27.760,0:00:30.880 on in the course 0:00:28.720,0:00:32.640 quantum mechanics is firmly rooted in 0:00:30.880,0:00:34.640 experimental observation 0:00:32.640,0:00:36.239 it wasn't that we thought physics needed 0:00:34.640,0:00:37.920 to get more magical so we started 0:00:36.239,0:00:38.640 inventing strange interpretations of 0:00:37.920,0:00:40.000 things 0:00:38.640,0:00:41.920 what happened is that we carried out a 0:00:40.000,0:00:44.000 set of experiments which revealed to us 0:00:41.920,0:00:45.520 that the world is indeed more magical 0:00:44.000,0:00:49.440 and we invented quantum mechanics in 0:00:45.520,0:00:52.800 order to explain those observations 0:00:49.440,0:00:56.480 so to begin with some pre-history 0:00:52.800,0:00:58.320 in 1756 flame tests were developed 0:00:56.480,0:00:59.840 in which you take a pure element and 0:00:58.320,0:01:00.640 heat is over a flame and the flame 0:00:59.840,0:01:01.760 changes colour 0:01:00.640,0:01:04.160 it's an experiment you may have done 0:01:01.760,0:01:04.160 yourself 0:01:04.479,0:01:09.200 the this is now understood to be a 0:01:06.960,0:01:11.840 result of the discrete nature of the 0:01:09.200,0:01:14.720 atomic levels 0:01:11.840,0:01:16.960 electrons occupy within atoms so the 0:01:14.720,0:01:17.360 word quantum in quantum mechanics refers 0:01:16.960,0:01:19.680 to 0:01:17.360,0:01:20.880 discrete or separate and it's what 0:01:19.680,0:01:22.960 happens when we 0:01:20.880,0:01:24.960 go from our large scale macroscopic 0:01:22.960,0:01:26.720 world which is seemingly continuous 0:01:24.960,0:01:28.000 and go down to the smallest scale we 0:01:26.720,0:01:31.920 kind of knew things had to get 0:01:28.000,0:01:33.439 discrete somehow. In 1801 Thomas Young 0:01:31.920,0:01:35.360 carried out an experiment which is now 0:01:33.439,0:01:37.439 called Young's slits 0:01:35.360,0:01:38.880 or the two-slit experiment in which he 0:01:37.439,0:01:40.880 took two 0:01:38.880,0:01:42.479 thin closely spaced slits in a piece of 0:01:40.880,0:01:44.880 card shone light through it 0:01:42.479,0:01:46.880 and measured a pattern on a screen and saw 0:01:44.880,0:01:48.640 that light is able to interfere 0:01:46.880,0:01:50.079 he took this as good evidence of the 0:01:48.640,0:01:51.840 wave nature of light 0:01:50.079,0:01:54.240 we'll take a look at that interference 0:01:51.840,0:01:57.040 pattern in another experiment 0:01:54.240,0:01:58.320 in 1850 atomic line spectra were measured 0:01:57.040,0:02:01.040 for the first time 0:01:58.320,0:02:02.479 we take a gas for pure element and 0:02:01.040,0:02:05.520 pass white light through it 0:02:02.479,0:02:07.040 and find that a set of discrete 0:02:05.520,0:02:09.520 frequencies will be removed from the 0:02:07.040,0:02:11.599 white light when we take that same gas 0:02:09.520,0:02:13.440 and heat it up we see the light that's 0:02:11.599,0:02:14.800 emitted from it is exactly that same set 0:02:13.440,0:02:16.640 of frequencies that were 0:02:14.800,0:02:18.319 absorbed from the light that was passed 0:02:16.640,0:02:19.040 through it we now know this to be 0:02:18.319,0:02:20.640 evidence 0:02:19.040,0:02:22.800 of the discrete nature of the atomic 0:02:20.640,0:02:26.239 levels once again 0:02:22.800,0:02:28.319 in 1887 the 0:02:26.239,0:02:29.680 photoelectric effect was measured by 0:02:28.319,0:02:31.599 Heinrich Hertz 0:02:29.680,0:02:32.959 so some gold leaf is taken and charged 0:02:31.599,0:02:34.560 electrically 0:02:32.959,0:02:36.000 two bits of gold leaf 0:02:34.560,0:02:38.080 would repel from one another 0:02:36.000,0:02:40.080 when light is shone onto the gold leaf we 0:02:38.080,0:02:41.440 see that the charge dissipates 0:02:40.080,0:02:43.200 the thing that was difficult to explain 0:02:41.440,0:02:45.200 classically was that 0:02:43.200,0:02:46.239 the frequency of the light being shone 0:02:45.200,0:02:47.840 onto the gold leaf 0:02:46.239,0:02:49.360 has to be above a certain threshold 0:02:47.840,0:02:51.440 frequency there was no classical 0:02:49.360,0:02:55.360 explanation of this 0:02:51.440,0:02:58.959 in 1897 J J Thompson and others 0:02:55.360,0:03:00.159 measured the behaviour of cathode rays 0:02:58.959,0:03:01.840 so initially you might have been 0:03:00.159,0:03:02.640 forgiven for thinking cathode rays were 0:03:01.840,0:03:04.720 something like 0:03:02.640,0:03:06.480 beams of light they were rays which 0:03:04.720,0:03:09.760 could light up 0:03:06.480,0:03:11.599 gases such as argon but unlike light 0:03:09.760,0:03:12.879 it was possible to bend these rays using 0:03:11.599,0:03:16.080 magnetic 0:03:12.879,0:03:17.599 fields so what this showed us is that 0:03:16.080,0:03:19.920 those rays were being carried 0:03:17.599,0:03:21.680 by massive particles hence their ability 0:03:19.920,0:03:22.879 to be deflected by magnetic fields and 0:03:21.680,0:03:24.799 accelerated 0:03:22.879,0:03:26.239 and so this was the first evidence for 0:03:24.799,0:03:29.360 subatomic particles 0:03:26.239,0:03:32.159 in this case electrons. 0:03:29.360,0:03:33.840 We've gone 0:03:32.159,0:03:36.000 through a set of experiments we've taken 0:03:33.840,0:03:38.319 this world that appears 0:03:36.000,0:03:39.440 on our everyday scales to be continuous 0:03:38.319,0:03:41.840 and we've observed that down on the 0:03:39.440,0:03:44.879 smaller scales that continuity 0:03:41.840,0:03:47.840 emerges out of small 0:03:44.879,0:03:47.840 sets of discrete things 0:03:48.080,0:03:51.599 in 1911 the Millikan experiment was 0:03:50.560,0:03:53.920 carried out in which 0:03:51.599,0:03:55.200 oil drops electrically charged could be 0:03:53.920,0:03:57.680 suspended using 0:03:55.200,0:03:59.519 electrostatic potentials by measuring 0:03:57.680,0:04:00.159 the required potential to levitate an 0:03:59.519,0:04:01.519 oil drop 0:04:00.159,0:04:03.599 it's possible to measure the charge 0:04:01.519,0:04:06.640 accurately and it was found 0:04:03.599,0:04:07.040 by Millikan that the charge always came 0:04:06.640,0:04:08.879 in 0:04:07.040,0:04:10.640 an integer multiple of some smallest 0:04:08.879,0:04:12.000 amount which we now understand to be the 0:04:10.640,0:04:14.239 charge of the electron 0:04:12.000,0:04:15.439 in 1913 the Rutherford experiment was 0:04:14.239,0:04:17.199 carried out also known as the 0:04:15.439,0:04:19.280 Geiger-Marsden experiment after the 0:04:17.199,0:04:21.600 people who really did most of the work 0:04:19.280,0:04:22.320 in which alpha particles were seen to 0:04:21.600,0:04:24.400 deflect 0:04:22.320,0:04:25.680 from atoms alpha particles we now 0:04:24.400,0:04:28.720 understand to be the nuclei 0:04:25.680,0:04:31.120 of helium four atoms so 0:04:28.720,0:04:33.120 occasionally these alpha particles would 0:04:31.120,0:04:34.560 deflect through more than 90 degrees 0:04:33.120,0:04:36.320 reflecting back 0:04:34.560,0:04:38.240 so this was evident that while the atom 0:04:36.320,0:04:40.080 is overall charged neutral 0:04:38.240,0:04:43.199 that charge is not evenly distributed 0:04:40.080,0:04:45.919 there's a small positively charged 0:04:43.199,0:04:46.800 nucleus and a negative charge around 0:04:45.919,0:04:48.240 that core 0:04:46.800,0:04:50.400 this led to a problem for classical 0:04:48.240,0:04:52.720 physics because if the 0:04:50.400,0:04:54.000 distribution is as described by the 0:04:52.720,0:04:55.680 Rutherford experiment 0:04:54.000,0:04:57.199 why don't the negative charges fall into 0:04:55.680,0:04:59.280 the positive charges to minimize their 0:04:57.199,0:05:00.479 energy 0:04:59.280,0:05:02.400 there's then a short break in the 0:05:00.479,0:05:05.199 experiments i'd like to mention owing 0:05:02.400,0:05:07.199 to the world war and world pandemic 0:05:05.199,0:05:10.960 and the next i want to mention is the 0:05:07.199,0:05:14.000 Compton scattering experiment in 1923. 0:05:10.960,0:05:17.759 This saw the scattering of light 0:05:14.000,0:05:20.080 by electrons and is used as evidence 0:05:17.759,0:05:22.479 that much like we've seen both the 0:05:20.080,0:05:23.919 particle and wave-like nature to light 0:05:22.479,0:05:25.600 you could see a particle and wave like 0:05:23.919,0:05:28.880 nature to electrons 0:05:25.600,0:05:29.840 in 1923 to 1927 the Davisson-Germer 0:05:28.880,0:05:32.000 experiment 0:05:29.840,0:05:33.840 saw interference patterns in the 0:05:32.000,0:05:34.560 electrons deflecting off the surface of 0:05:33.840,0:05:36.960 nickel 0:05:34.560,0:05:39.520 so this was further evidence for the 0:05:36.960,0:05:43.680 fact that particles such as the electron 0:05:39.520,0:05:43.680 can have wave-like characteristics 0:05:44.000,0:05:48.639 so that's a set of experiments that 0:05:47.120,0:05:51.280 does not exhaust the set of experiments 0:05:48.639,0:05:52.000 that require quantum explanations rather 0:05:51.280,0:05:54.400 than classical 0:05:52.000,0:05:55.840 but it's some of the key examples 0:05:54.400,0:05:57.120 coincident with this of course were 0:05:55.840,0:05:59.120 theoretical developments. 0:05:57.120,0:06:00.960 Quantum mechanics has a very nice 0:05:59.120,0:06:02.479 history of theory and experiment working 0:06:00.960,0:06:05.280 together 0:06:02.479,0:06:06.720 so the pre-history i think one of the 0:06:05.280,0:06:08.479 most important things to mention before 0:06:06.720,0:06:11.440 the development of quantum mechanics 0:06:08.479,0:06:13.120 was Maxwell's equations you the unified 0:06:11.440,0:06:14.319 theory of electromagnetism 0:06:13.120,0:06:16.800 and in particular the prediction of 0:06:14.319,0:06:18.800 electromagnetic waves so 0:06:16.800,0:06:20.120 Maxwell's equations were written down 0:06:18.800,0:06:23.360 in around 0:06:20.120,0:06:25.600 1865 something like that 0:06:23.360,0:06:27.360 and while coming before special 0:06:25.600,0:06:29.759 relativity and quantum mechanics 0:06:27.360,0:06:31.280 they're completely compatible with both 0:06:29.759,0:06:33.759 so 0:06:31.280,0:06:35.039 what this means is that light emits both 0:06:33.759,0:06:36.639 a classical description 0:06:35.039,0:06:37.919 which is wholly accurate provided 0:06:36.639,0:06:39.440 there are no interactions so 0:06:37.919,0:06:41.280 non-interacting light can be described 0:06:39.440,0:06:43.039 completely classically in terms of waves 0:06:41.280,0:06:44.400 but it can also be described completely 0:06:43.039,0:06:46.720 quantum mechanically in terms of 0:06:44.400,0:06:48.960 particles which we now call photons 0:06:46.720,0:06:51.120 so this is a useful trick that'll allow 0:06:48.960,0:06:53.280 us to do various experiments 0:06:51.120,0:06:56.240 in our own home of a quantum mechanical 0:06:53.280,0:06:56.240 nature using light 0:06:56.720,0:07:02.960 in 1900 Lord Rayleigh 0:07:00.080,0:07:03.840 and shortly after James Jeans in 1905 0:07:02.960,0:07:06.000 with Rayleigh 0:07:03.840,0:07:08.000 developed a theory of the spectral 0:07:06.000,0:07:11.120 radiance of black bodies 0:07:08.000,0:07:14.560 so what this means is the 0:07:11.120,0:07:16.560 set of frequencies coming off a body 0:07:14.560,0:07:18.880 at a given temperature 0:07:16.560,0:07:19.759 so all bodies emit black body 0:07:18.880,0:07:21.599 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 0:07:23.360,0:07:26.400 black things such as pure carbon for 0:07:25.680,0:07:28.639 example 0:07:26.400,0:07:31.520 being very good absorbers give very 0:07:28.639,0:07:33.680 clear black body radiation spectra 0:07:31.520,0:07:36.160 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 0:07:45.440,0:07:47.599 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 0:07:55.759,0:08:00.240 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). 1 00:00:01,680 --> 00:00:06,640 hello the behaviour of particles in 2 00:00:04,560 --> 00:00:10,240 quantum mechanics is governed by what's 3 00:00:06,640 --> 00:00:10,240 called the schrodinger equation 4 00:00:10,559 --> 00:00:14,480 written here in its time-dependent form 5 00:00:13,280 --> 00:00:18,160 so we sometimes call this 6 00:00:14,480 --> 00:00:18,160 the time-dependent schrodinger equation 7 00:00:18,400 --> 00:00:27,199 usually abbreviated to TDSE for short 8 00:00:23,039 --> 00:00:30,480 it was written down in 1925 9 00:00:27,199 --> 00:00:32,320 by this man Erwin Schrodginer as you can 10 00:00:30,480 --> 00:00:32,880 see a very well-dressed gentleman as 11 00:00:32,320 --> 00:00:35,120 they 12 00:00:32,880 --> 00:00:36,000 were back in the 1920s but you know 13 00:00:35,120 --> 00:00:37,520 what, it's the 14 00:00:36,000 --> 00:00:40,000 20s again so let's see if we can't 15 00:00:37,520 --> 00:00:42,640 conjure up a bit of that 1920s chic 16 00:00:40,000 --> 00:00:42,640 in this video 17 00:00:42,960 --> 00:00:48,960 it worked. Magic! OK. 18 00:00:46,399 --> 00:00:50,480 So the schrodinger equation when 19 00:00:48,960 --> 00:00:52,320 schrodinger originally wrote it down 20 00:00:50,480 --> 00:00:55,280 was intended to be a classical 21 00:00:52,320 --> 00:00:57,520 description of classical waves 22 00:00:55,280 --> 00:00:59,280 this quantity psi which we call the wave 23 00:00:57,520 --> 00:01:00,960 function was thought of 24 00:00:59,280 --> 00:01:02,960 as something like the wave function 25 00:01:00,960 --> 00:01:05,519 which would describe say water waves 26 00:01:02,960 --> 00:01:06,640 in which case it would take some value 27 00:01:05,519 --> 00:01:07,600 which would tell us about the height of 28 00:01:06,640 --> 00:01:11,040 the water wave 29 00:01:07,600 --> 00:01:13,119 as a function of position and time 30 00:01:11,040 --> 00:01:14,080 in fact as we'll see while there is a 31 00:01:13,119 --> 00:01:15,439 good description of 32 00:01:14,080 --> 00:01:17,520 the wave function as something like a 33 00:01:15,439 --> 00:01:19,439 classical wave, in fact quantum mechanics 34 00:01:17,520 --> 00:01:23,040 goes beyond classical physics as we now 35 00:01:19,439 --> 00:01:24,720 of course know. So before we 36 00:01:23,040 --> 00:01:26,159 proceed let me just make a couple of 37 00:01:24,720 --> 00:01:28,479 notational points 38 00:01:26,159 --> 00:01:30,079 when i write d(psi)/dt like this, this 39 00:01:28,479 --> 00:01:30,640 is a partial derivative with respect to 40 00:01:30,079 --> 00:01:32,960 time 41 00:01:30,640 --> 00:01:34,079 holding all spatial coordinates constant 42 00:01:32,960 --> 00:01:36,479 so we can also denote this in the 43 00:01:34,079 --> 00:01:39,119 following way 44 00:01:36,479 --> 00:01:41,200 so we denote this 45 00:01:39,119 --> 00:01:43,200 stating explicitly that 46 00:01:41,200 --> 00:01:44,720 the positions are all constant 47 00:01:43,200 --> 00:01:46,159 sometimes i'll denote this 48 00:01:44,720 --> 00:01:47,759 in a simpler manner in the following 49 00:01:46,159 --> 00:01:49,920 form 50 00:01:47,759 --> 00:01:50,960 where partial subscript t is just 51 00:01:49,920 --> 00:01:54,320 shorthand for 52 00:01:50,960 --> 00:01:54,640 d by dt acting on psi and sometimes i'll 53 00:01:54,320 --> 00:01:57,759 use 54 00:01:54,640 --> 00:02:00,159 newton's notation psi dot 55 00:01:57,759 --> 00:02:01,600 similarly a partial derivative with 56 00:02:00,159 --> 00:02:02,960 respect to x 57 00:02:01,600 --> 00:02:04,719 implies that we're holding the other 58 00:02:02,960 --> 00:02:08,479 variables constant so time 59 00:02:04,719 --> 00:02:10,640 y and z which can again be abbreviated 60 00:02:08,479 --> 00:02:12,319 just partial subscript x acting on 61 00:02:10,640 --> 00:02:13,840 psi 62 00:02:12,319 --> 00:02:15,920 and sometimes we'll write this as psi 63 00:02:13,840 --> 00:02:17,280 prime 64 00:02:15,920 --> 00:02:19,120 where this last notation will only 65 00:02:17,280 --> 00:02:21,040 really be used in one dimension where 66 00:02:19,120 --> 00:02:23,840 it's unambiguously a derivative with 67 00:02:21,040 --> 00:02:26,080 respect to x 68 00:02:23,840 --> 00:02:26,959 so there are a couple of key 69 00:02:26,080 --> 00:02:28,480 assumptions we're going to make 70 00:02:26,959 --> 00:02:31,200 throughout this course let's just wipe 71 00:02:28,480 --> 00:02:32,640 this board off and write them down 72 00:02:31,200 --> 00:02:35,519 when i say wipe the board off of course 73 00:02:32,640 --> 00:02:38,800 i mean magically clear the board 74 00:02:35,519 --> 00:02:41,120 so let's write down some key assumptions 75 00:02:38,800 --> 00:02:41,920 the first is that the schrodinger 76 00:02:41,120 --> 00:02:44,000 equation 77 00:02:41,920 --> 00:02:45,599 is non-relativistic. It's going to 78 00:02:44,000 --> 00:02:46,239 describe the behavior of some massive 79 00:02:45,599 --> 00:02:48,080 particle 80 00:02:46,239 --> 00:02:50,000 in the non-relativistic limit. There are 81 00:02:48,080 --> 00:02:51,599 relativistic extensions to it 82 00:02:50,000 --> 00:02:54,160 but we're not going to consider those in 83 00:02:51,599 --> 00:02:54,160 this course 84 00:02:54,239 --> 00:02:58,080 the second is that we're only going to 85 00:02:56,239 --> 00:03:00,400 describe single particles. 86 00:02:58,080 --> 00:03:01,840 Again there are extensions to the 87 00:03:00,400 --> 00:03:03,519 schrodinger equation which will allow it 88 00:03:01,840 --> 00:03:04,720 to deal with multiple particles 89 00:03:03,519 --> 00:03:06,560 but that's not going to be the focus of 90 00:03:04,720 --> 00:03:08,159 this course. The wave function psi 91 00:03:06,560 --> 00:03:10,560 always governs the behavior of a single 92 00:03:08,159 --> 00:03:10,560 particle 93 00:03:10,879 --> 00:03:14,640 okay so let's take a closer look at 94 00:03:13,440 --> 00:03:18,000 what's going on in the schrodinger 95 00:03:14,640 --> 00:03:19,280 equation let's clear the board again 96 00:03:18,000 --> 00:03:21,440 so let's have the schrodinger equation 97 00:03:19,280 --> 00:03:23,280 back 98 00:03:21,440 --> 00:03:25,519 so we'll look at properties of the wave 99 00:03:23,280 --> 00:03:27,280 function psi in future videos 100 00:03:25,519 --> 00:03:30,560 but for now let's just note that it's a 101 00:03:27,280 --> 00:03:32,560 complex-valued function 102 00:03:30,560 --> 00:03:33,680 and it has out the front of it an 103 00:03:32,560 --> 00:03:37,599 arbitrary 104 00:03:33,680 --> 00:03:39,680 global complex phase. 105 00:03:37,599 --> 00:03:41,120 It's a mathematical redundancy built 106 00:03:39,680 --> 00:03:42,640 into the the wave function 107 00:03:41,120 --> 00:03:44,560 it has this global phase which can be 108 00:03:42,640 --> 00:03:46,560 changed arbitrarily. 111 00:03:47,440 --> 00:03:51,680 But the relative phase between two 112 00:03:50,480 --> 00:03:53,360 different wave functions 113 00:03:51,680 --> 00:03:54,959 is important and we will see that later 114 00:03:53,360 --> 00:03:57,120 on in the course 115 00:03:54,959 --> 00:03:58,400 but for now there is this redundancy 116 00:03:57,120 --> 00:04:01,280 built into it. 118 00:04:01,280 --> 00:04:05,120 We've got 119 00:04:02,720 --> 00:04:07,680 the imaginary unit here i 120 00:04:05,120 --> 00:04:09,120 square root minus one. Imaginary 121 00:04:07,680 --> 00:04:10,400 numbers play an important role in 122 00:04:09,120 --> 00:04:12,159 quantum mechanics. 123 00:04:10,400 --> 00:04:14,560 hbar here is the reduced planck's 124 00:04:12,159 --> 00:04:18,799 constant h over 2 pi 125 00:04:14,560 --> 00:04:18,799 with units of energy multiplied by time 126 00:04:18,959 --> 00:04:22,160 and then we have this quantity here H 127 00:04:21,600 --> 00:04:23,919 which 128 00:04:22,160 --> 00:04:26,240 is called the hamiltonian let's write 129 00:04:23,919 --> 00:04:29,360 that down 130 00:04:26,240 --> 00:04:32,880 so h is defined as follows 131 00:04:29,360 --> 00:04:34,720 it's a differential 132 00:04:32,880 --> 00:04:35,280 operator we have this nabla squared in 133 00:04:34,720 --> 00:04:39,680 here 134 00:04:35,280 --> 00:04:39,680 which okay we can expand as follows 135 00:04:40,080 --> 00:04:43,759 so it's just the partial derivative 136 00:04:42,560 --> 00:04:45,680 with respect to x 137 00:04:43,759 --> 00:04:47,520 squared plus that of y squared and that 138 00:04:45,680 --> 00:04:49,840 of z squared remembering the notation 139 00:04:47,520 --> 00:04:51,520 from the last board and if this looks 140 00:04:49,840 --> 00:04:54,800 confusing just remember this is always 141 00:04:51,520 --> 00:04:57,600 acting on psi so this term here 142 00:04:54,800 --> 00:04:57,919 (d/dx)^2 acting on psi is really just 143 00:04:57,600 --> 00:05:01,199 psi 144 00:04:57,919 --> 00:05:05,039 double prime. Okay. 145 00:05:01,199 --> 00:05:06,800 So it's called the Hamiltonian 146 00:05:05,039 --> 00:05:08,320 and it's what's called an energy 147 00:05:06,800 --> 00:05:11,440 operator for the system 148 00:05:08,320 --> 00:05:13,759 so the hamiltonian acting on psi 149 00:05:11,440 --> 00:05:15,199 is going to return the energy of the 150 00:05:13,759 --> 00:05:18,639 system as we'll see 151 00:05:15,199 --> 00:05:21,440 in a second in general 152 00:05:18,639 --> 00:05:23,039 specifying the hamiltonian specifies the 153 00:05:21,440 --> 00:05:23,759 entire quantum problem we want to solve 154 00:05:23,039 --> 00:05:24,880 so when we 155 00:05:23,759 --> 00:05:26,479 write down a quantum mechanical 156 00:05:24,880 --> 00:05:27,919 description of a system we just need to 157 00:05:26,479 --> 00:05:29,919 write down the hamiltonian 158 00:05:27,919 --> 00:05:32,240 and in fact this first term is fixed 159 00:05:29,919 --> 00:05:34,000 what we what need to specify is the 160 00:05:32,240 --> 00:05:35,600 potential of the system which is much 161 00:05:34,000 --> 00:05:37,280 like what you do in classical mechanics 162 00:05:35,600 --> 00:05:39,680 as well 163 00:05:37,280 --> 00:05:41,600 so the potential we will assume in 164 00:05:39,680 --> 00:05:44,320 this course is time independent 165 00:05:41,600 --> 00:05:44,880 it doesn't need to be but we won't 166 00:05:44,320 --> 00:05:48,320 consider 167 00:05:44,880 --> 00:05:51,440 time dependent potentials here. So 168 00:05:48,320 --> 00:05:52,000 let's take a look at attempts to 169 00:05:51,440 --> 00:05:55,120 deal with 170 00:05:52,000 --> 00:05:55,120 how to solve this equation 171 00:05:55,199 --> 00:05:59,680 so the time-dependent schrodinger 172 00:05:57,120 --> 00:05:59,680 equation 173 00:05:59,840 --> 00:06:03,600 where i've written it with psi dot this 174 00:06:01,440 --> 00:06:06,800 time is a separable 175 00:06:03,600 --> 00:06:08,479 equation what we mean by this 176 00:06:06,800 --> 00:06:10,560 is that we can substitute the 177 00:06:08,479 --> 00:06:13,600 following anzatz 178 00:06:10,560 --> 00:06:17,280 we can say that psi(x,t) 179 00:06:13,600 --> 00:06:18,960 is equal to: 180 00:06:17,280 --> 00:06:21,759 -- and let's treat it in three 181 00:06:18,960 --> 00:06:25,039 dimensions in general -- is equal to 182 00:06:21,759 --> 00:06:28,080 phi(x) only 183 00:06:25,039 --> 00:06:29,759 and T(t) only 184 00:06:28,080 --> 00:06:31,280 so these are two separate functions one 185 00:06:29,759 --> 00:06:32,880 of which is only a function of time 186 00:06:31,280 --> 00:06:35,440 and one of which is only a function of 187 00:06:32,880 --> 00:06:36,080 position when we substitute 188 00:06:35,440 --> 00:06:37,919 that in 189 00:06:36,080 --> 00:06:40,240 we see that the equation reduces to the 190 00:06:37,919 --> 00:06:42,960 following form 191 00:06:40,240 --> 00:06:44,000 where the phi has pulled through the 192 00:06:42,960 --> 00:06:45,840 time derivative 193 00:06:44,000 --> 00:06:48,240 because it's not a function of time and 194 00:06:45,840 --> 00:06:51,440 this is a partial derivative 195 00:06:48,240 --> 00:06:52,479 the partial derivative of T(t) with respect 196 00:06:51,440 --> 00:06:54,000 to time 197 00:06:52,479 --> 00:06:55,520 is actually a total derivative because 198 00:06:54,000 --> 00:06:57,440 it's only a function of time 199 00:06:55,520 --> 00:06:59,360 and over here the hamiltonian because the 200 00:06:57,440 --> 00:07:01,520 potential is time independent 201 00:06:59,360 --> 00:07:03,599 hamiltonian only acts on the spatial 202 00:07:01,520 --> 00:07:06,240 coordinates so we can pull through the t 203 00:07:03,599 --> 00:07:09,199 over here rearranging slightly we get 204 00:07:06,240 --> 00:07:09,199 two different equations 205 00:07:10,479 --> 00:07:15,039 one purely in terms of t and one purely 206 00:07:13,120 --> 00:07:17,520 in terms of x 207 00:07:15,039 --> 00:07:19,360 so we've turned what was a partial 208 00:07:17,520 --> 00:07:20,080 differential equation in terms of both x 209 00:07:19,360 --> 00:07:22,479 and t 210 00:07:20,080 --> 00:07:23,840 into two separate equations one is an 211 00:07:22,479 --> 00:07:24,400 ordinary differential equation in terms 212 00:07:23,840 --> 00:07:25,599 of time 213 00:07:24,400 --> 00:07:27,440 and the other is potentially still a 214 00:07:25,599 --> 00:07:29,120 partial differential equation in terms 215 00:07:27,440 --> 00:07:31,360 of positions 216 00:07:29,120 --> 00:07:32,160 so the fact that these two are equal for 217 00:07:31,360 --> 00:07:34,080 all times 218 00:07:32,160 --> 00:07:35,759 and all positions means they must both 219 00:07:34,080 --> 00:07:36,880 be equal to the same constant 220 00:07:35,759 --> 00:07:39,360 and we're going to suggestively call 221 00:07:36,880 --> 00:07:41,599 that constant E 222 00:07:39,360 --> 00:07:42,800 suggestive because it should remind us 223 00:07:41,599 --> 00:07:45,280 of an energy 224 00:07:42,800 --> 00:07:47,599 so let's number these equations let's 225 00:07:45,280 --> 00:07:50,400 call this one (i) 226 00:07:47,599 --> 00:07:50,400 and this one (ii) 227 00:07:50,960 --> 00:07:54,479 we'll move this up to the top of the 228 00:07:53,120 --> 00:07:56,639 board and we'll deal with (i) and (ii) 229 00:07:54,479 --> 00:07:59,440 separately 230 00:07:56,639 --> 00:08:02,319 so (i) simply rearranges 231 00:07:59,440 --> 00:08:02,319 to the following form 232 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 234 00:08:11,120 --> 00:08:15,520 called the time 235 00:08:12,080 --> 00:08:17,120 independent schrodinger equation because 236 00:08:15,520 --> 00:08:19,120 phi here is now only a function of 237 00:08:17,120 --> 00:08:22,639 positions not of time. 238 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 240 00:08:24,639 --> 00:08:28,720 in general it's impossible to solve 241 00:08:27,120 --> 00:08:30,560 remember the hamiltonian here contains 242 00:08:28,720 --> 00:08:32,159 all the real information in the problem 243 00:08:30,560 --> 00:08:34,080 in the form of the potential 244 00:08:32,159 --> 00:08:36,320 so in general we need to solve this 245 00:08:34,080 --> 00:08:38,080 differential equation 246 00:08:36,320 --> 00:08:39,440 in terms of boundary conditions which we 247 00:08:38,080 --> 00:08:42,000 specify and 248 00:08:39,440 --> 00:08:42,800 you can see that it's an eigenvalue 249 00:08:42,000 --> 00:08:45,440 equation 250 00:08:42,800 --> 00:08:45,839 in that we need to solve this for 251 00:08:45,440 --> 00:08:48,080 both 252 00:08:45,839 --> 00:08:49,519 the eigenfunctions phi(x) and the 253 00:08:48,080 --> 00:08:50,560 eigenenergies E 254 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 279 00:09:50,800 --> 00:09:54,560 okay thank you for your time it must have been 280 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 282 00:09:55,440 --> 00:10:00,080 I think I'll turn back into my normal 283 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 0:05:26.000,0:05:30.479 we can pull the p through the x at the 0:05:28.080,0:05:31.680 expense of adding in this extra term 0:05:30.479,0:05:35.520 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 0:05:45.440,0:05:49.520 the p operator followed by the x 0:05:46.800,0:05:51.280 operator all followed by the x operator 0:05:49.520,0:05:52.720 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 0:06:00.240,0:06:05.199 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 0:06:10.160,0:06:13.440 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 0:06:16.880,0:06:22.160 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 0:06:28.720,0:06:32.400 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 0:06:47.440,0:06:49.599 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. 0:00:00.399,0:00:03.120 hello in this video we're going to take 0:00:02.320,0:00:05.359 a look at 0:00:03.120,0:00:06.960 infinite dimensional hilbert spaces or 0:00:05.359,0:00:08.080 rather we're going to reassess what 0:00:06.960,0:00:09.120 we've already done and see that we've 0:00:08.080,0:00:10.800 actually already been working with 0:00:09.120,0:00:13.120 infinite dimensional hilbert spaces 0:00:10.800,0:00:14.559 when we've worked with functions so 0:00:13.120,0:00:16.480 we've had two schemes that we've worked 0:00:14.559,0:00:18.800 with the first is wave mechanics 0:00:16.480,0:00:20.560 and second is matrix mechanics in the 0:00:18.800,0:00:22.880 second we've had quantities like vectors 0:00:20.560,0:00:24.480 and matrices 0:00:22.880,0:00:26.480 and we're familiar with how to 0:00:24.480,0:00:28.480 manipulate these things when 0:00:26.480,0:00:30.000 the dimension of the space in which 0:00:28.480,0:00:32.239 we're working has a finite number of 0:00:30.000,0:00:35.360 dimensions 0:00:32.239,0:00:38.480 in the case that we extend our space to 0:00:35.360,0:00:42.320 have an infinite number of dimensions 0:00:38.480,0:00:45.039 the objects just become the following: 0:00:42.320,0:00:46.800 vectors become functions and matrices 0:00:45.039,0:00:48.719 become differential operators 0:00:46.800,0:00:50.800 and in general we can use the 0:00:48.719,0:00:54.160 overarching terms 0:00:50.800,0:00:56.320 'states', referring either to vectors in 0:00:54.160,0:00:58.399 finite dimensional spaces or functions 0:00:56.320,0:01:00.480 in infinite dimensional spaces, and 0:00:58.399,0:01:04.080 'operators' referring to either matrices 0:01:00.480,0:01:05.680 or differential operators so 0:01:04.080,0:01:07.280 this might seem a little bit abstract 0:01:05.680,0:01:09.200 thinking of a function say 0:01:07.280,0:01:10.640 as an infinite dimensional vector but 0:01:09.200,0:01:11.520 there's a couple of reasons that it's 0:01:10.640,0:01:13.520 quite natural 0:01:11.520,0:01:17.040 the first is if we think in terms of how 0:01:13.520,0:01:18.880 a computer would display a function 0:01:17.040,0:01:21.439 that is we would have to work with 0:01:18.880,0:01:24.560 the discrete set of positions if the 0:01:21.439,0:01:27.840 function is defined in position space 0:01:24.560,0:01:29.600 which would be stored as a vector 0:01:27.840,0:01:31.280 of different points and for each of 0:01:29.600,0:01:32.320 those points we'd have a value of our 0:01:31.280,0:01:34.880 function 0:01:32.320,0:01:36.159 now if we want to approximate a 0:01:34.880,0:01:39.200 smooth function 0:01:36.159,0:01:40.159 taking a value in the in a real 0:01:39.200,0:01:42.000 domain 0:01:40.159,0:01:43.200 then we'd have to try and make the 0:01:42.000,0:01:46.479 spacing of these points 0:01:43.200,0:01:48.479 smaller and smaller something like this 0:01:46.479,0:01:50.320 and as we take the limit of the number 0:01:48.479,0:01:51.520 of points here going to infinity and the 0:01:50.320,0:01:52.880 spacing between each one being 0:01:51.520,0:01:56.000 infinitesimal 0:01:52.880,0:01:57.439 we develop a smooth function 0:01:56.000,0:01:59.360 but of course the computer only ever 0:01:57.439,0:02:01.200 works with a discrete set of 0:01:59.360,0:02:02.880 points along the real line it can't 0:02:01.200,0:02:05.920 store 0:02:02.880,0:02:07.119 an infinite set of numbers it only has a 0:02:05.920,0:02:09.360 finite memory 0:02:07.119,0:02:10.800 so in this sense you see that a function 0:02:09.360,0:02:12.239 really is naturally an infinite 0:02:10.800,0:02:14.160 dimensional vector 0:02:12.239,0:02:15.680 because it has to take a value for each 0:02:14.160,0:02:19.760 of an infinite number of 0:02:15.680,0:02:22.160 real numbers of x another way to see 0:02:19.760,0:02:23.040 in terms of matrices why this is is that 0:02:22.160,0:02:24.239 if you think of 0:02:23.040,0:02:26.400 we've been looking at eigenvalue 0:02:24.239,0:02:29.440 equations from matrices 0:02:26.400,0:02:30.640 and for an n by n matrix the 0:02:29.440,0:02:32.640 eigenvalues are found by a 0:02:30.640,0:02:35.280 characteristic polynomial 0:02:32.640,0:02:36.000 and that characteristic polynomial 0:02:35.280,0:02:38.720 has 0:02:36.000,0:02:40.720 the form some coefficient times x 0:02:38.720,0:02:42.319 plus another coefficient times x squared 0:02:40.720,0:02:45.200 plus another coefficient times x cubed 0:02:42.319,0:02:48.319 and so on up to x to the power of n 0:02:45.200,0:02:49.040 so it's an nth order polynomial if we 0:02:48.319,0:02:50.959 take the 0:02:49.040,0:02:52.400 limit of the matrix becoming infinitely 0:02:50.959,0:02:54.879 large the polynomial 0:02:52.400,0:02:55.760 that describes the characteristic 0:02:54.879,0:02:58.400 equation 0:02:55.760,0:02:59.920 has to have an infinite number of terms 0:02:58.400,0:03:00.879 an infinite number of different powers 0:02:59.920,0:03:03.519 of x 0:03:00.879,0:03:04.959 but if you write a polynomial which has 0:03:03.519,0:03:06.640 an infinite number of powers of x in it 0:03:04.959,0:03:08.400 like that what you've really written is 0:03:06.640,0:03:09.599 a Taylor series which describes an 0:03:08.400,0:03:12.239 arbitrary function 0:03:09.599,0:03:14.080 and we can always expand a function of x 0:03:12.239,0:03:14.560 as a taylor series like that or there's 0:03:14.080,0:03:16.000 a 0:03:14.560,0:03:18.400 large set of functions for which we can 0:03:16.000,0:03:21.440 expand it as a Taylor series so 0:03:18.400,0:03:23.120 your matrix then 0:03:21.440,0:03:25.120 the characteristic polynomial stops 0:03:23.120,0:03:26.879 being an nth order polynomial 0:03:25.120,0:03:29.440 becomes an infinite order polynomial 0:03:26.879,0:03:30.879 which is really just a function 0:03:29.440,0:03:33.120 so it's quite natural to think of it in 0:03:30.879,0:03:35.040 this way 0:03:33.120,0:03:36.879 so in our finite dimensional vector 0:03:35.040,0:03:40.720 spaces we can define 0:03:36.879,0:03:43.519 orthonormal bases defined as follows 0:03:40.720,0:03:44.720 that is sets of normalized vectors such 0:03:43.519,0:03:46.400 that the inner product of the vector 0:03:44.720,0:03:47.920 with itself will give one. This is the 0:03:46.400,0:03:50.080 kronecker delta defined to be 0:03:47.920,0:03:51.440 one when i equals j or zero when i 0:03:50.080,0:03:53.439 doesn't equal j 0:03:51.440,0:03:56.000 so if e_i and e_j are the same this is 0:03:53.439,0:03:58.080 length one and if e_i and e_j 0:03:56.000,0:03:59.200 are sorry i and j are different then 0:03:58.080,0:04:01.200 it's equal to zero 0:03:59.200,0:04:02.400 and this defines an orthonormal basis of 0:04:01.200,0:04:04.239 vectors 0:04:02.400,0:04:06.959 in the infinite dimensional space the 0:04:04.239,0:04:10.000 equivalent to this is as follows 0:04:06.959,0:04:13.280 we define an orthonormal basis 0:04:10.000,0:04:15.920 of states x 0:04:13.280,0:04:16.320 which are our position states and we say 0:04:15.920,0:04:17.919 that 0:04:16.320,0:04:19.600 if you have the inner product of x with 0:04:17.919,0:04:22.240 y that's equal to 0:04:19.600,0:04:24.720 the dirac delta function of x minus y 0:04:22.240,0:04:27.040 this is in one dimension of space 0:04:24.720,0:04:28.880 so remember the dirac delta function is 0:04:27.040,0:04:32.160 really like a continuum limit of 0:04:28.880,0:04:34.960 the kronecker delta 0:04:32.160,0:04:36.880 it's equal to zero if x doesn't equal y 0:04:34.960,0:04:38.560 and it equals infinity when x does equal 0:04:36.880,0:04:39.120 y but in such a way that's an integral 0:04:38.560,0:04:41.199 over it 0:04:39.120,0:04:42.800 will give the value one so it's the 0:04:41.199,0:04:46.960 natural generalization 0:04:42.800,0:04:49.600 so whereas we have a finite number of 0:04:46.960,0:04:50.479 vectors which span a finite dimensional 0:04:49.600,0:04:52.639 vector space 0:04:50.479,0:04:54.000 we must have an infinite number of these 0:04:52.639,0:04:55.360 position vectors 0:04:54.000,0:04:57.040 but again you can just think of this in 0:04:55.360,0:04:58.479 terms of how a computer would store 0:04:57.040,0:05:00.080 a set of real numbers it would really 0:04:58.479,0:05:00.639 approximate them as a finite number of 0:05:00.080,0:05:02.320 points 0:05:00.639,0:05:04.000 so then it would be a finite dimensional 0:05:02.320,0:05:04.560 vector space and these would be the 0:05:04.000,0:05:08.000 points 0:05:04.560,0:05:08.000 along that that space 0:05:09.120,0:05:12.240 so another relation we have in the 0:05:10.400,0:05:13.680 finite dimensional case is that the 0:05:12.240,0:05:15.440 identity matrix 0:05:13.680,0:05:17.360 which we can denote with this sort of 0:05:15.440,0:05:19.759 blackboard bold 0:05:17.360,0:05:21.280 one is equal to the sum from i equals 0:05:19.759,0:05:22.000 one to n where this is the dimension of 0:05:21.280,0:05:25.520 the space 0:05:22.000,0:05:28.880 of the outer products of the 0:05:25.520,0:05:30.720 normalized basis vectors in the 0:05:28.880,0:05:33.520 infinite dimensional case the 0:05:30.720,0:05:33.520 equivalent of this 0:05:33.600,0:05:38.720 so the identity now which is really an 0:05:36.400,0:05:42.000 identity operator 0:05:38.720,0:05:45.600 is equal to the outer product of the 0:05:42.000,0:05:48.160 position states integrated now 0:05:45.600,0:05:48.960 over x from minus infinity to 0:05:48.160,0:05:51.199 infinity 0:05:48.960,0:05:53.199 so the integral is replacing the sum 0:05:51.199,0:05:54.880 because we've taken the limit 0:05:53.199,0:05:57.440 of the number of dimensions going to 0:05:54.880,0:05:57.440 infinity 0:05:58.639,0:06:02.800 in the finite dimensional case let's 0:06:01.520,0:06:04.160 keep all the finite dimensional stuff to 0:06:02.800,0:06:06.479 the side of the line 0:06:04.160,0:06:07.440 we can express any vector in the space 0:06:06.479,0:06:09.280 as a sum 0:06:07.440,0:06:11.039 from y equals one to n then the size of 0:06:09.280,0:06:14.319 the space the dimension of the space 0:06:11.039,0:06:15.360 multiplying the basis vectors of the 0:06:14.319,0:06:18.479 space 0:06:15.360,0:06:20.240 by some coefficient of v_i and the 0:06:18.479,0:06:20.960 coefficients v_i are given by the inner 0:06:20.240,0:06:24.720 product 0:06:20.960,0:06:25.759 of the basis vector with the vector v 0:06:24.720,0:06:28.800 the equivalent to the infinite 0:06:25.759,0:06:31.199 dimensional case is this 0:06:28.800,0:06:32.639 so let's denote rather than choosing v 0:06:31.199,0:06:33.039 for our finite dimensional vectors we'll 0:06:32.639,0:06:34.800 choose 0:06:33.039,0:06:36.240 f for our infinite dimensional ones 0:06:34.800,0:06:39.199 because these are going to be functions 0:06:36.240,0:06:42.639 and we can write them now as 0:06:39.199,0:06:45.680 replace the sum with the integral 0:06:42.639,0:06:46.800 so it's some function of x multiplying 0:06:45.680,0:06:49.120 the basis states 0:06:46.800,0:06:50.319 x and this function of x is defined to 0:06:49.120,0:06:53.360 be 0:06:50.319,0:06:56.880 the inner product of the x state 0:06:53.360,0:06:59.120 so the basis vector with the function f 0:06:56.880,0:07:00.319 so this takes a bit of getting used to 0:06:59.120,0:07:01.520 but what we're saying here is that 0:07:00.319,0:07:04.080 what we usually call 0:07:01.520,0:07:04.720 our functions of x are really some more 0:07:04.080,0:07:06.319 abstract 0:07:04.720,0:07:10.400 concept they're an infinite dimensional 0:07:06.319,0:07:12.319 vector projected into the x basis. 0:07:10.400,0:07:13.599 So what other bases could they be 0:07:12.319,0:07:15.120 projected into? 0:07:13.599,0:07:16.639 Well actually they can be projected 0:07:15.120,0:07:18.400 into other bases 0:07:16.639,0:07:19.520 because we can write the function as a 0:07:18.400,0:07:20.960 function of anything it doesn't have to 0:07:19.520,0:07:22.720 be a function of position 0:07:20.960,0:07:24.400 an example that we've actually seen 0:07:22.720,0:07:26.319 already is that 0:07:24.400,0:07:28.639 you can write your functions in momentum 0:07:26.319,0:07:30.960 space and what you would do 0:07:28.639,0:07:32.720 mathematically to write a function in 0:07:30.960,0:07:34.400 position as a function of momentum is 0:07:32.720,0:07:35.919 carry out a fourier transform 0:07:34.400,0:07:38.560 and that's built in naturally into this 0:07:35.919,0:07:40.319 structure so 0:07:38.560,0:07:41.759 the function is actually a slightly more 0:07:40.319,0:07:43.680 abstract entity than 0:07:41.759,0:07:46.080 the thing that's a function of x the 0:07:43.680,0:07:50.000 function could be a function of x or p 0:07:46.080,0:07:52.639 or of anything so this ket notation 0:07:50.000,0:07:53.520 indicates that the sort of true nature 0:07:52.639,0:07:55.120 of the function 0:07:53.520,0:07:57.199 before it's projected into a particular 0:07:55.120,0:07:58.960 basis we'll see a concrete example of 0:07:57.199,0:08:02.479 this in a second 0:07:58.960,0:08:04.080 another key generalization we have 0:08:02.479,0:08:06.960 is this 0:08:04.080,0:08:07.680 so we have an inner product between our 0:08:06.960,0:08:10.160 vectors 0:08:07.680,0:08:10.800 and remember it's a complex vector space 0:08:10.160,0:08:14.560 so 0:08:10.800,0:08:15.680 writing the ket backwards like 0:08:14.560,0:08:18.960 this -- the bra -- 0:08:15.680,0:08:21.440 is the hermitian conjugate 0:08:18.960,0:08:22.000 and remember a Hilbert space which we're 0:08:21.440,0:08:24.720 working with 0:08:22.000,0:08:26.479 is defined to be a linear vector space 0:08:24.720,0:08:28.000 but including an inner product 0:08:26.479,0:08:29.599 and having all the vectors that we're 0:08:28.000,0:08:32.880 referring to be normalizable 0:08:29.599,0:08:34.880 so of finite length and this generalizes 0:08:32.880,0:08:36.719 quite nicely in the basis of functions 0:08:34.880,0:08:39.519 as follows 0:08:36.719,0:08:40.959 so just as we can write the inner 0:08:39.519,0:08:42.640 product of two vectors u and v 0:08:40.959,0:08:44.080 like this we can write the inner product 0:08:42.640,0:08:47.040 of two functions 0:08:44.080,0:08:47.680 f and g in exactly the same way but to 0:08:47.040,0:08:50.880 make it look 0:08:47.680,0:08:52.839 more standard we can 0:08:50.880,0:08:54.720 use the trick of inserting the 0:08:52.839,0:08:55.839 identity okay so if we insert the 0:08:54.720,0:08:58.160 identity between f and g 0:08:55.839,0:08:59.200 we have the following we can always just 0:08:58.160,0:09:01.519 put an identity 0:08:59.200,0:09:02.640 between two vectors but then we use this 0:09:01.519,0:09:05.680 identity up here 0:09:02.640,0:09:07.360 to rewrite this as follows so an 0:09:05.680,0:09:09.839 integral over dx 0:09:07.360,0:09:11.279 and we've inserted the complete set of 0:09:09.839,0:09:13.680 outer products here 0:09:11.279,0:09:14.800 but then according to our definition up 0:09:13.680,0:09:16.959 here 0:09:14.800,0:09:19.519 x inner products with g like this is 0:09:16.959,0:09:22.800 just g of x 0:09:19.519,0:09:24.320 and f inner product x well that has to 0:09:22.800,0:09:25.440 be the hermitian conjugate sorry the 0:09:24.320,0:09:27.600 complex conjugate 0:09:25.440,0:09:29.200 because this is just a complex number 0:09:27.600,0:09:30.959 now because it's an inner product 0:09:29.200,0:09:33.040 so this has to be the complex conjugate 0:09:30.959,0:09:36.560 of x inner product f 0:09:33.040,0:09:38.720 so it's as follows 0:09:36.560,0:09:40.240 so the equivalent to the inner product 0:09:38.720,0:09:42.640 in the vector space 0:09:40.240,0:09:44.160 is the integral from minus infinity to 0:09:42.640,0:09:47.600 infinity over x 0:09:44.160,0:09:49.040 of f*(x)g(x) 0:09:47.600,0:09:50.880 okay so this is the inner product 0:09:49.040,0:09:52.240 defined for functions in fact we've used 0:09:50.880,0:09:55.839 that implicitly when we're working with 0:09:52.240,0:09:55.839 wave mechanics earlier on in the course 0:09:56.000,0:09:59.920 so the most important example of an 0:09:58.640,0:10:01.360 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 0:10:06.000,0:10:09.440 which we've been using in the wave 0:10:07.040,0:10:12.480 mechanics part of the course 0:10:09.440,0:10:16.320 in terms of matrix mechanics is just 0:10:12.480,0:10:18.160 the ket psi projected into the x basis 0:10:16.320,0:10:21.600 but we could equally well have projected 0:10:18.160,0:10:24.000 it into the momentum basis 0:10:21.600,0:10:24.880 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