List of Definitions assumed from Quantum Mechanics

The canonical commutation relation: \(\left[\hat{x},\hat{p}\right]=i\hbar\hat{\mathbb{I}}\)
Dirac notation: the notation \(|\psi\rangle\) for complex vectors. Also called bra-ket notation, with \(\langle\phi|\) the ‘bra’, \(|\psi\rangle\) the ‘ket’, and \(\langle\phi|\psi\rangle\) a bracket.
Expectation value: \(\langle\hat{A}\rangle=\langle\psi|\hat{A}|\psi\rangle\). The mean value of an operator measured by a given state.
First quantization: a wave-like description of quantum objects: \(\psi\left(x\right)\).
The Hamiltonian: the energy operator (assumed time independent in this course). \(\hat{H}=\hat{p}^{2}/2m+\hat{V}\), or \(\hat{H}\psi\left(x\right)=-\hbar^{2}\psi''/2m+V\left(x\right)\psi\).
The Heisenberg picture: the description of quantum states as time independent, and operators as time dependent.
The Heisenberg uncertainty principle: \(\sigma_{\hat{A}}\sigma_{\hat{B}}\ge\frac{1}{2}\left|\left\langle \left[\hat{A},\hat{B}\right]\right\rangle \right|\) where \(\sigma_{\hat{A}}\) denotes the standard deviation of operator \(\hat{A}\).
Hilbert space: a vector space with an inner product and square-normalisable vectors
Hermiticity: \(\hat{A}=\hat{A}^{\dagger}\) where \(\hat{A}^{\dagger}=\hat{A}^{*T}\). For differential operators: \(\int^{\infty}_{-\infty}\varphi\left(x\right)^{*}\left(\hat{A}\psi\left(x\right)\right)\text{d}x=\int^{\infty}_{-\infty}\left(\hat{A}\varphi\left(x\right)\right)^{*}\psi\left(x\right)\text{d}x\).
Ladder operators: an operator which raises or lowers the quantum number of a state it acts on. Also called creation/annihilation operators or raising and lowering operators.
Normalisation: the prefactor on a wavefunction ensuring that the total probability to find the particle is one.
The number operator: in the harmonic oscillator, the operator whose eigenstates are the energy eigenstates and whose eigenvalues are the level of the state.
Operators: objects which act on states to give states (either the same state, or a different one). In finite dimensional Hilbert spaces these are simply matrices acting on vectors to give vectors. In infinite dimensional Hilbert spaces they can be differential operators acting on functions to give functions.
Orthonormality: orthogonal and normalised. If a set of states is orthonormal the inner product of any state with itself is 1 and the inner product between any two different states is zero.
The probability density \(\rho\): integrated over a region of space, this gives the probability to find the particle in that region. \(\rho\left(x,t\right)\text{d}x=\left|\psi\left(x,t\right)\right|^{2}\text{d}x\) is the probability to find the particle between \(x\) and \(x+\text{d}x\) at time \(t\).
The probability current density \(j\): the current density associated with a flow of probability: \(\boldsymbol{j}\left(\boldsymbol{x},t\right)=\frac{i\hbar}{2m}\left\{ \psi\nabla\psi^{*}-\psi^{*}\nabla\psi\right\}\).
The probability amplitude: the complex number associated to each point in space by the wavefunction \(\psi\).
Quantum numbers: eigenvalues of operators which commute with the Hamiltonian; expectation values which do not change in time.
The Schrodinger picture: the description of quantum states as time dependent, and operators as time independent.
Second quantisation: a particle-like description of quantum objects in terms of ladder operators.
Stationary states: energy eigenstates. So called as their probability densities are time independent.
Superposition: summing solutions to the TDSE to get a new solution to the TDSE
The time dependent schroedinger equation: \(i\hbar\partial_{t}|\psi\rangle=\hat{H}|\psi\rangle\), or in the position basis \(i\hbar\dot{\psi}=\hat{H}\psi\). Abbreviated TDSE.
The time independent schroedinger equation: \(\hat{H}|\psi\rangle=E|\psi\rangle\), or in the position basis \(\hat{H}\psi\left(x\right)=E\psi\left(x\right)\). Abbreviated TISE.
The wavefunction \(\psi\): a function which assigns a complex number to each point in space. The modulus square is the probability density \(\rho\).