This is a table of contents for my notes on quantum mechanics. The posts are arranged in an order that might be found in a textbook.

- Euler-Lagrange equations for particle and field theories
- Coupled masses on springs - a solution using matrix diagonalization
- Coupled masses on springs - properties of the propagator
- Lagrangians for harmonic oscillators
- Lagrangian for classical electromagnetism
- Lagrangian for the two-body problem
- Hamiltonian formalism and Legendre transformations
- Hamiltonians for harmonic oscillators
- Hamiltonian for the two-body problem
- Cyclic coordinates and Poisson brackets
- Poisson brackets, commutators and Jacobi identity
- Commutator of two hermitian operators

- Schrödinger equation
- Time-independent Schrödinger equation
- Real solutions to the Schrödinger equation
- Even and odd solutions to the Schrödinger equation
- Schrödinger equation - minimum energy
- No degenerate solutions to the Schrödinger equation in one dimension
- Wave function as a probability
- Probability current
- Ehrenfest's theorem
- Commutators - a few theorems
- Classical limit of quantum mechanics - Ehrenfest's theorem
- Born's conditions on the wave function
- Postulates of quantum mechanics - states and measurements
- Postulates of quantum mechanics - Schrödinger equation and propagators
- Momentum eigenfunctions and eigenvalues
- Momentum - eigenvalues and normalization
- Uncertainty principle
- Uncertainty principle for position and energy
- Uncertainty principle - condition for minimum uncertainty
- Uncertainty principle - rates of change of operators
- Uncertainty principle in three dimensions
- Extended uncertainty principle
- Energy-time uncertainty relation
- Energy-time uncertainty principle - example
- Position operator - eigenfunctions
- Non-denumerable basis - position and momentum states
- Wave function with arbitrary initial state
- Reflectionless potential
- Schrödinger equation and propagators
- Time-dependent propagators

- Infinite square well
- Infinite square well - combination of two lowest states
- Infinite square well - minimum energy
- Infinite square well - uncertainty principle
- Infinite square well - centered coordinates
- Infinite square well - change in well size
- Quantum revival time in infinite square well
- Infinite square well - numerical solution
- Energy-time uncertainty principle - infinite square well
- Energy-time uncertainty - an alternative definition
- Virial theorem
- Infinite square well - momentum space wave functions
- Infinite square well in three dimensions
- Infinite spherical well - spherical Bessel functions
- Infinite spherical well - numerical solutions

- Harmonic oscillator - algebraic solution
- Harmonic oscillator - three lowest stationary states
- Harmonic oscillator - position, momentum and energy
- Harmonic oscillator - raising and lowering operator calculations
- Creation and annihilation operators in the harmonic oscillator - a few theorems
- Creation and annihilation operators for the 3-d harmonic oscillator
- Harmonic oscillator - probability of being outside classical region
- Harmonic oscillator - asymptotic solution
- Harmonic oscillator - series solution
- Hermite polynomials
- Harmonic oscillator - Hermite polynomials
- Hermite polynomials - generation
- Hermite polynomials - the Rodrigues formula
- Hermite polynomials - recursion relations
- Harmonic oscillator - example starting state
- Harmonic oscillator - Schrödinger's exact solution
- Momentum space - harmonic oscillator
- Harmonic oscillator ground state - numerical solution
- Harmonic oscillator excited states - numerical solution
- Two-dimensional harmonic oscillator
- Harmonic oscillator in 3-d - rectangular coordinates
- Harmonic oscillator in 2-d and 3-d, and in polar and spherical coordinates
- Harmonic oscillator in 3-d spherical coordinates
- Two-dimensional harmonic oscillator - comparison with rectangular coordinates
- Hamiltonian matrix elements
- Harmonic oscillator - matrix elements
- Harmonic oscillator - coherent states
- Harmonic oscillator - first order perturbation
- Feynman-Hellmann theorem and the harmonic oscillator

- Free particle
- Free particle as a wave packet
- Uncertainty principle - visualization with Fourier series
- Fourier transforms and Plancherel's theorem
- Free particle - probability current
- Free particle as Gaussian wave packet
- Free particle - travelling wave packet
- Free particle as moving Gaussian wave packet
- Free particle propagator
- Energy-time uncertainty principle - Gaussian free particle
- Free particle in momentum space
- Particle on a circular wire

- Dirac delta function
- Dirac delta function - Fourier transform
- Delta-function well - bound state
- Delta function well bound state - uncertainty principle
- Delta-function well - scattering
- Delta function well as limit of finite square well
- Double delta function well
- Double delta function well - scattering states

- Finite square well - bound states, even wave functions
- Finite square well - bound states, odd wave functions
- Finite square well - normalization
- Finite square well - scattering
- Finite square well - numerical solution
- Finite square barrier - scattering
- Finite step potential - scattering
- Finite drop potential
- Hybrid infinite-finite square well
- Finite spherical well

- Vector spaces
- Subspaces and direct sums
- Span, linear independence and basis
- Vector spaces and linear independence - some examples
- Vector spaces - number of dimensions
- Linear maps, linear operators and commutators
- Matrix representation of linear operators
- Rotation matrices - matrix elements
- Linear operators - change of basis, trace and determinant
- Null space, range, injectivity and surjectivity
- Inverses of linear operators
- Eigenvalues and eigenvectors
- Eigenvalues and eigenvectors - examples
- Eigenvalues and eigenvectors of the 2-d rotation operator
- Inner products and Hilbert spaces
- Orthonormal basis and orthogonal complement
- Projection operators
- Linear functionals and adjoint operators
- Hermitian operators
- Hermitian operators - equivalence of conditions
- Hermitian operators - a few theorems
- Hermitian operators - common eigenfunctions implies they commute
- Hermitian conjugate (adjoint) of an operator
- Hermitian matrices - example with 4 matrices
- Degenerate eigenvalues and Gram-Schmidt orthogonalization
- Gram-Schmidt orthogonalization - a couple of examples
- Anti-hermitian operators
- Unitary operators
- Unitary matrices - some examples
- Unitary operators - active and passive transformations of an operator
- Diagonalization of matrices
- Normal operators
- Spectral theorem for normal operators
- Determinant and trace of normal operators
- Simultaneous diagonalization of hermitian matrices
- Functions of hermitian operators
- Hadamard's lemma
- Baker-Campbell-Hausdorff formula
- Projection operators
- Projection operators - formal treatment
- Finite vector spaces - matrix elements
- Spectral decomposition of operators
- Hamiltonian for three-state system
- Hamiltonian and observables in three-state system
- Derivatives of the delta function
- Differential operators - matrix elements and hermiticity
- Differential operator - eigenvalues and eigenstates

- Invariance of Euler-Lagrange and Hamilton's equations under canonical transformations
- Conditions for a transformation to be canonical
- Cyclic coordinates and Poisson brackets
- Passive, regular and active transformations
- Translational invariance in quantum mechanics
- Translation operator from passive transformations
- Translational invariance and conservation of momentum
- Finite transformations - correspondence between classical and quantum
- Translation invariance in two dimensions
- Correspondence between classical and quantum transformations
- Parity transformations
- Time translation and conservation of energy
- Unitary transformations and the Heisenberg picture
- Time reversal, antiunitary operators and Wigner's theorem

- De Broglie waves
- Rotational invariance in two dimensions
- Rotational transformations using passive transformations
- Rotations through a finite angle - use of polar coordinates
- Eigenvalues of two-dimensional angular momentum
- Angular momentum - probabilities of eigenvalues in two dimensions
- Radially symmetric potentials, angular momentum and centrifugal force
- Combining translations and rotations
- Angular momentum in three dimensions
- Finite rotations about an arbitrary axis in three dimensions
- Vector operators - transformation under rotation
- Rotation of a vector wave function
- Angular momentum - commutators
- Angular momentum - eigenvalues
- Associated Legendre functions
- Associated Legendre functions - orthogonality
- Angular momentum - raising and lowering operators
- Angular momentum - commutators with position and momentum
- Angular momentum - eigenfunctions
- Angular momentum as a generator of rotations

- Schrödinger equation in three dimensions - spherical harmonics
- Spherical harmonics - examples
- Spherical harmonics - more examples
- Spherical harmonic at the top of the ladder
- Spherical harmonic using the raising operator
- Spherical harmonics - normalization
- Legendre polynomials - orthogonality
- Schrödinger equation in three dimensions - the radial equation
- Self-adjoint differential equations
- Virial theorem in 3-d
- Probability current in 3-d
- Electromagnetic force law in quantum mechanics

- Hydrogen atom - radial equation
- Laguerre polynomials - normalization
- Hydrogen atom - series solution and Bohr energy levels
- Energy levels of hydrogen - Bohr's semi-classical derivation
- Hydrogen atom - radial function examples
- Hydrogen atom - wave function examples
- Hydrogen atom - mean radius of electron position
- Hydrogen atom - spectrum
- Hydrogen atom - complete wave function
- Hydrogen atom - radial functions for large L
- Hydrogen atom - powers of the momentum operator
- Feynman-Hellmann theorem - hydrogen atom mean values
- Kramers's relation for averages of radial powers in hydrogen
- Kramers's relation - application to hydrogen mean values

- Electron as a classical spinning sphere
- Spin - introduction
- Spin one-half and the Pauli spin matrices
- Eigenspinors of the Pauli spin matrices
- Spin - statistical calculations
- Spin - expectation values of components
- Spin - the x and y components
- Spin one-half - spin components
- Spin one-half along an arbitrary direction
- Spin one-half particle in a magnetic field
- Spin 1
- Spin three-halves
- Spin matrices - general case

- Angular momentum - adding 2 spins
- Angular momentum - addition and Clebsch-Gordan coefficients
- Angular momentum - commutators of added spins
- Angular momentum - adding spins in arbitrary directions
- Clebsch-Gordan coefficients for higher spin
- Angular momentum - adding 3 spins

- Schrödinger equation for 2 particles - separation of variables
- Identical particles - fermions and bosons
- Infinite square well - 2 particle systems
- Exchange force - infinite square well
- Fermions and bosons - n-particle systems
- Fermion wave functions; the Slater determinant
- Helium atom
- Helium atom - parahelium and orthohelium
- Helium atom - electron-electron interaction
- Periodic table
- Hund's rules

- Electron gas - a crude model of a solid
- Degeneracy pressure in a solid
- Periodic potentials - Bloch's theorem and the band structure of solids
- Band structure of solids - numerical solution
- Band structure of solids - negative energies
- Band structure of solids - degeneracy of states

- Statistical mechanics in quantum theory - counting states
- Statistical mechanics in quantum theory - counting states, general case
- Statistical mechanics in quantum theory - most probable state
- Statistical mechanics in quantum theory - most probable state for fermions
- Statistical mechanics in quantum theory - Bose condensation
- Statistical mechanics in quantum theory - 3-d harmonic oscillator
- Blackbody radiation
- Stefan-Boltzmann law
- Electron pressure in a white dwarf star
- Electron pressure in a neutron star

- First order non-degenerate perturbation theory
- Harmonic oscillator - first order perturbation
- Creation and annihilation operators in a perturbed harmonic oscillator
- Second order non-degenerate perturbation theory
- Degenerate perturbation theory - two states
- Perturbing a particle on a circular wire
- Perturbation theory for higher-level degenerate systems
- Degenerate perturbation in 3 state system
- Feynman-Hellmann theorem and the harmonic oscillator
- Feynman-Hellmann theorem - hydrogen atom mean values
- Perturbation theory and the variational principle

- Fine structure constant
- Fine structure of hydrogen - relativistic correction
- Fine structure of hydrogen - spin-orbit coupling
- Fine structure of hydrogen - spin-orbit eigenstates and final formula
- Hyperfine splitting and the 21 cm line of hydrogen

- Zeeman effect - strength of magnetic field
- Zeeman effect - weak field
- Zeeman effect - strong field
- Zeeman effect - the n=2 line in hydrogen
- Zeeman effect for l=0
- Zeeman effect - degenerate perturbation theory for n = 2
- Zeeman effect for n = 3 - weak field
- Zeeman effect for n = 3 - strong field
- Zeeman effect for n = 3 - general case
- Stark effect in hydrogen for n = 1 and n = 2
- Stark effect in hydrogen for n = 3 - hydrogen wave functions in Maple
- Perturbing the wave function (Stark effect and proton electric dipole moment)
- Stark effect - tunnelling probability

- Variational principle in quantum mechanics
- Variational principle and the harmonic oscillator
- Variational principle and the harmonic oscillator - 2 lowest levels
- Variational principle and harmonic oscillator - a more general trial function
- Variational principle and the delta function well
- Variational principle and the first excited state
- Perturbation theory and the variational principle
- Helium atom using the variational principle
- Rubber band helium
- Hydrogen molecule ion
- Hydrogen molecule ion - different trial function
- Hydrogen molecule ion - oscillation of the protons
- Variational principle and the hydrogen atom
- Variational principle and the hydrogen ion - two parameters
- Variational principle and the Yukawa potential
- Variational principle with a two-state hamiltonian
- Quantum dots

- Adiabatic approximation in quantum mechanics
- Phases in the adiabatic approximation
- Geometric phase is always zero for real wave functions

- Rutherford scattering
- Quantum scattering - scattering amplitude and differential cross section
- Quantum scattering - partial wave analysis
- Phase shift in one-dimensional scattering
- Partial waves in three dimensions - hard sphere scattering
- Integral form of the Schrödinger equation
- Integral form of the Schrödinger equation - ground state of hydrogen
- First Born approximation - soft-sphere scattering
- Born approximation for a spherical delta function shell
- Scattering from the Yukawa potential
- Impulse approximation in scattering theory
- Second order Born approximation in scattering theory
- Green's function for one dimensional Schrödinger equation
- Born approximation in one dimension
- Born approximation of delta function well and finite square well
- Optical theorem

- WKB approximation
- WKB approximation - tunneling
- Alpha decay using the WKB approximation
- Airy functions and the bouncing electron
- WKB approximation - turning points
- WKB approximation - analysis of the overlap region near a turning point
- WKB approximation at a turning point with decreasing potential
- WKB approximation of the harmonic oscillator
- WKB approximation for a barrier with sloping sides
- WKB approximation and the radial equation
- WKB approximation and the hydrogen atom
- WKB approximation of a double potential well - turning points
- WKB approximation of double-well potential - wave functions
- Stark effect - tunnelling probability
- Half life of a beer can

- Time dependent Schrödinger equation - two-state systems
- Time-dependent Schrödinger equation - switching a perturbation on and off
- Time-dependent perturbation theory - iterative solution
- Time-dependent perturbation theory - general two-state solution
- Sinusoidal perturbations in time
- Time-dependent perturbation theory for a multi-level system
- Magnetic resonance

- Stimulated emission of radiation - lasers
- Stimulated emission of radiation at high frequencies
- Spontaneous emission - Einstein's argument
- Spontaneous emission rates for the hydrogen atom
- Selection rules for spontaneous emission of radiation
- Selection rules in spontaneous emission - transition between spherically symmetric states not allowed
- Forbidden transitions in the harmonic oscillator and hydrogen
- Spontaneous emission rates for hydrogen - general solution