Quantum mechanics - Table of contents

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.

Review of classical mechanics

• 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

• 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
• 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

• 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
• 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

Delta function well

• 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 & barrier

• 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

Linear algebra in quantum mechanics

• 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

Transformations and symmetries

• 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

Rotations and angular momentum

• 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

• 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

• 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

Spin in non-relativistic quantum mechanics

• 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

Addition of angular momenta

• 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

Multiple particles

• 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

Quantum solids

• 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

• 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

Time-independent perturbation theory

• 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 of hydrogen

• 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 and Stark effects

• 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

• 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

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

Scattering

• 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
• 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 perturbation theory

• 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

Emission of radiation

• 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