- Adding a constant to the potential introduces a phase factor
- Adiabatic approximation in quantum mechanics
- Adiabatic approximation: higher order corrections
- Airy functions and the bouncing electron
- Alpha decay using the WKB approximation
- Angular equation – alternative solution
- Angular momentum: adding 2 spins
- Angular momentum: adding 3 spins
- Angular momentum: adding spins in arbitrary directions
- Angular momentum: addition and Clebsch-Gordan coefficients
- Angular momentum – commutators
- Angular momentum commutators in hydrogen
- Angular momentum – commutators with position and momentum
- Angular momentum: commutators of added spins
- Angular momentum – eigenfunctions
- Angular momentum – eigenvalues
- Angular momentum as a generator of rotations
- Angular momentum – raising and lowering operators
- Angular momentum: restriction to integer values
- Angular momentum and torque
- Anti-hermitian operators
- Atomic wavefunctions: symmetrization
- Average and standard deviation
- Band structure of solids: degeneracy of states
- Band structure of solids: numerical solution
- Band structure of solids: negative energies
- Berry’s phase: definition and value for a spin-1 particle in a magnetic field
- Blackbody radiation
- Born approximation in one dimension
- Born approximation of delta function well and finite square well
- Born approximation for a spherical delta function shell
- Bosons in the infinite square well
- Buffon’s needle: estimating pi
- Bulk modulus in the electron gas model
- Clebsch-Gordan coefficients for higher spin
- Commutators: a few theorems
- Complex exponentials and trig functions
- Continuous probability distribution: needle on a pivot
- Degeneracy pressure in a solid
- Degenerate eigenvalues and Gram-Schmidt orthogonalization
- Degenerate perturbation theory: two states
- Degenerate perturbation in 3 state system
- Degenerate solutions don’t exist in one dimension
- Delta-function well – bound state
- Delta function well: bound state – uncertainty principle
- Delta-function well – scattering
- Delta function well: statistics
- Delta function potential – moving delta function
- Delta function well as limit of finite square well
- Delta function – Fourier transform
- Delta function in time perturbation
- Determinate states
- Dirac delta function
- Dirac delta function in three dimensions
- Dirac delta function – simple examples
- Double delta function well
- Double delta function well – scattering states
- Dysprosium electron configuration
- Earth-Sun system as a quantum atom
- Electrodynamics in quantum mechanics: gauge transformations
- Electromagnetic force law in quantum mechanics
- Electromagnetism in quantum mechanics: example
- Electron as a classical spinning sphere
- Electron gas: a crude model of a solid
- Electron gas in a 2-d infinite square well
- Electron in a precessing magnetic field
- Electron pressure in a neutron star
- Electron pressure in a white dwarf star
- Energy levels of hydrogen: Bohr’s semi-classical derivation
- Energy states: bound and scattering states
- Energy & wave functions: a few theorems
- Energy-time uncertainty relation
- Energy-time uncertainty: an alternative definition
- Energy-time uncertainty principle – example
- Energy-time uncertainty principle: Gaussian free particle
- Energy-time uncertainty principle: infinite square well
- Exchange force: harmonic oscillator
- Exchange force: infinite square well
- Extended uncertainty principle
- Fermions and bosons: counting states
- Fermions and bosons: n-particle systems
- A few statistics on the first 25 digits of pi
- Feynman-Hellmann theorem and the harmonic oscillator
- Feynman-Hellmann theorem: hydrogen atom mean values
- Fine structure constant
- Fine structure of hydrogen: Dirac formula
- Fine structure of hydrogen: relativistic correction
- Fine structure of hydrogen: spin-orbit coupling
- Fine structure of hydrogen: spin-orbit eigenstates and final formula
- Fine structure of a spectral line in hydrogen
- Finite drop potential
- Finite spherical well
- Finite square barrier – scattering
- Finite square well: bound states & even wave functions
- Finite square well: bound states & odd wave functions
- Finite square well – normalization
- Finite square well – numerical solution
- Finite square well – scattering
- Finite step potential – scattering
- First Born approximation: soft-sphere scattering
- First order non-degenerate perturbation theory
- Forbidden transitions in the harmonic oscillator and hydrogen
- Forced harmonic oscillator: exact solution and adiabatic approximation
- The free particle
- Free particle in momentum space
- The free particle as a wave packet
- Free particle: Gaussian wave packet
- Free particle – travelling wave packet
- The free particle: probability current
- Fusion with a muon-deuteron system
- Gaussian distribution
- Geometric phase is always zero for real wave functions
- Green’s function for one dimensional Schrödinger equation
- Half-harmonic oscillator
- Half life of a beer can
- Hamiltonian matrix elements
- Hamiltonian in two-level system
- Hamiltonian for three-state system
- Hamiltonian and observables in three-state system
- Harmonic oscillator: algebraic solution; raising and lowering operators
- Harmonic oscillator: algebraic normalization of raising and lowering operators
- Harmonic oscillator – asymptotic solution
- Harmonic oscillator – change in spring constant
- Harmonic oscillator: coherent states
- Harmonic oscillator in an electric field
- Harmonic oscillator – probability of being outside classical region
- Harmonic oscillator ground state – numerical solution
- Harmonic oscillator excited states – numerical solution
- Harmonic oscillator: first order perturbation
- Harmonic oscillator – Hermite polynomials
- Harmonic oscillator: matrix elements
- Harmonic oscillator – mixed initial state
- Harmonic oscillator: mixture of two lowest states
- Harmonic oscillator – position, momentum and energy
- Harmonic oscillator – raising and lowering operator calculations
- Harmonic oscillator: relativistic correction
- Harmonic oscillator: Schrödinger’s exact solution
- Harmonic oscillator – example starting state
- Harmonic oscillator – series solution
- Harmonic oscillator: statistics
- Harmonic oscillator – summary
- Harmonic oscillator – three lowest stationary states
- Harmonic oscillator in 3-d – rectangular coordinates
- Harmonic oscillator in 3-d: spherical coordinates
- Helium atom
- Helium atom: electron-electron interaction
- Helium atom: parahelium and orthohelium
- Helium atom using the variational principle
- Hermite polynomials – recursion relations
- Hermite polynomials – generation
- Hermite polynomials – the Rodrigues formula
- Hermitian conjugate of an operator
- Hermitian operators
- Hermitian operators: common eigenfunctions implies they commute
- Hermitian operators – equivalence of conditions
- Hermitian operators – a few theorems
- Hermitian operators: periodic function
- Hilbert space – power functions
- Hund’s rules
- Hybrid infinite-finite square well
- Hydrogen atom: coincident spectral lines
- Hydrogen atom: combined position and spin state
- Hydrogen atom in a crystal lattice
- Hydrogen atom – Laguerre polynomials example
- Hydrogen atom – mean radius of electron position
- Hydrogen atom – mixed initial state and mean potential energy
- Hydrogen atom – most probable distance of electron
- Hydrogen atom: powers of the momentum operator
- Hydrogen atom: probability of finding electron inside the nucleus
- Hydrogen atom -radial equation
- Hydrogen atom – radial function examples
- Hydrogen atom: radial functions for large l
- Hydrogen atom – series solution and Bohr energy levels
- Hydrogen atom – spectrum
- Hydrogen atom – wave function example 1
- Hydrogen atom – complete wave function & example 2
- Hydrogen atom: wave function example 3
- Hydrogen-like atoms
- Hydrogen molecule ion
- Hydrogen molecule ion – different trial function
- Hydrogen molecule ion – oscillation of the protons
- Hyperfine splitting and the 21 cm line of hydrogen
- Hyperfine splitting in deuterium
- Identical particles: fermions and bosons
- Impulse approximation in scattering theory
- Infinite spherical well – numerical solutions
- Infinite spherical well – spherical Bessel functions
- The infinite square well (particle in a box)
- Infinite square well – average energy
- Infinite square well – centered coordinates
- Infinite square well – change in well size
- Infinite square well – combination of two lowest states
- Infinite square well – cubic sine initial state
- Infinite square well – minimum energy
- Infinite square well: momentum
- Infinite square well: momentum space wave functions
- Infinite square well – numerical solution
- Infinite square well – particle in left half
- Infinite square well – phase difference
- Infinite square well in three dimensions
- Infinite square well – triangular initial state
- Infinite square well with triangular initial state using delta function
- Infinite square well: 2 particle systems
- Infinite square well – uncertainty principle
- Infinite square well with delta function barrier
- Infinite square well with variable delta function barrier: ground state energy
- Infinite square well with variable delta function barrier: location of the particle
- Inner product of two wave functions is constant in time
- Integral form of the Schrödinger equation
- Integral form of the Schrödinger equation: ground state of hydrogen
- Kramers’s relation for averages of radial powers in hydrogen
- Kramers’s relation: application to hydrogen mean values
- Laguerre polynomials – normalization
- Legendre polynomials: generation by Gram-Schmidt process
- Magnetic resonance
- Matrix elements: example
- Momentum: eigenvalues and normalization
- Momentum space representation of finite wave function
- Momentum space: harmonic oscillator
- Momentum space: mean position
- Momentum space: another example
- Momentum space in 3-d
- The need for quantum theory
- Optical theorem
- Partial waves in three dimensions: hard sphere scattering
- Particle on a circular wire
- Periodic potentials: Bloch’s theorem and the band structure of solids
- Periodic table
- Perturbation due to finite size of the proton in hydrogen
- Perturbation theory for higher-level degenerate systems
- Perturbation theory and the variational principle
- Perturbing the 3-d harmonic oscillator
- Perturbation of 3-d square well
- Perturbing a particle on a circular wire
- Perturbing the wave function (Stark effect and proton electric dipole moment)
- Phase shift in one-dimensional scattering
- Phase shift in the spherical delta function shell
- Phases in the adiabatic approximation
- Phases in the adiabatic theorem: delta function well
- Plancherel’s theorem
- Position and momentum
- Position operator: eigenfunctions
- Probability current
- Probability current in 3-d
- Projection operators
- Quantum versus classical mechanics in solids and gases
- Quantum dots
- Quantum revival time
- Quantum scattering: partial wave analysis
- Quantum scattering: scattering amplitude and differential cross section
- Reflectionless potential
- Rigid rotor in quantum mechanics
- Rubber band helium
- Rutherford scattering
- Scattering matrix
- Scattering from the Yukawa potential
- The Schrödinger equation
- The Schrödinger equation: the motivation
- Schrödinger equation – a few theorems
- Schrödinger equation – minimum energy
- Schrödinger equation in three dimensions – spherical harmonics
- Schrödinger equation in three dimensions – radial equation
- Schrödinger equation for 2 particles – separation of variables
- Second order Born approximation in scattering theory
- Second order non-degenerate perturbation theory
- Selection rules for spontaneous emission of radiation
- Selection rules in spontaneous emission: transition between spherically symmetric states not allowed
- Self-adjoint differential equations
- Sequential measurements
- Sinusoidal perturbations in time
- Spectral decomposition of operators
- Spherical harmonics – examples
- Spherical harmonics – more examples
- Spherical harmonics: normalization
- Spherical harmonic at the top of the ladder
- Spherical harmonic using the raising operator
- Spin – expectation values of components
- Spin – introduction
- Spin ½
- Spin 1/2 along an arbitrary direction
- Spin 1/2: minimum uncertainty
- Spin 1/2 particle in a magnetic field
- Spin 1/2 particle in time-varying magnetic field
- Spin 1/2: spin components
- Spin 1
- Spin 3/2
- Spin matrices: general case
- Spin and quarks
- Spin – statistical calculations
- Spin: the y component
- Spontaneous emission: Einstein’s argument
- Spontaneous emission rates for the hydrogen atom
- Spontaneous emission rates for hydrogen: general solution
- Spontaneous emission from n=3 to n=1 in hydrogen
- Spontaneous emission from the zero point field
- Stark effect in hydrogen for n = 1 and n = 2
- Stark effect in hydrogen for n = 3; hydrogen wave functions in Maple
- Stark effect: tunnelling probability
- Statistical mechanics in quantum theory: 3-d harmonic oscillator
- Statistical mechanics in quantum theory: Bose condensation
- Statistical mechanics in quantum theory: counting states
- Statistical mechanics in quantum theory: counting boson states
- Statistical mechanics in quantum theory: counting states, general case
- Statistical mechanics in quantum theory: energy probabilities
- Statistical mechanics in quantum theory: most probable state
- Statistical mechanics in quantum theory: most probable state for fermions
- Stefan-Boltzmann law
- Stimulated emission of radiation: lasers
- Stimulated emission of radiation at high frequencies
- Time-dependent perturbation theory: iterative solution
- Time-dependent perturbation theory: general two-state solution
- Time-dependent perturbation theory: switching a perturbation on and off
- Time-dependent perturbation theory for a multi-level system
- Time-dependent perturbation theory: transition probabilities in a multi-state system
- Time-dependent perturbation uniform in space
- Time-dependent perturbation of the infinite square well
- Time dependent Schrödinger equation: two-state systems
- Time-dependent Schrödinger equation: switching a perturbation on and off
- The time-independent Schrödinger equation
- The time-independent Schrödinger equation – general solutions
- Transfer matrix
- Translations in space and time
- Triangular wave function: probabilities
- Uncertainty principle
- Uncertainty principle: an example
- Uncertainty principle – examples
- Uncertainty principle: condition for minimum uncertainty
- Uncertainty principle: rates of change of operators
- Uncertainty principle in three dimensions
- Unstable particles: a crude model
- Van der Waals interaction
- The variational principle in quantum mechanics
- Variational principle and the delta function well
- Variational principle and the electron in a magnetic field
- Variational principle and the first excited state
- Variational principle and the harmonic oscillator
- Variational principle and the harmonic oscillator – 2
- Variational principle and harmonic oscillator: a more general trial function
- Variational principle and the hydrogen atom
- Variational principle and the hydrogen ion: two parameters
- Variational principle with a two-state hamiltonian
- Variational principle and the Yukawa potential
- Vector spaces and Hilbert space
- Vibrational states in a diatomic molecule (HCl)
- Virial theorem
- Virial theorem in 3-d
- WKB approximation
- WKB approximation – alternative derivation
- WKB approximation – analysis of the overlap region near a turning point
- WKB approximation for a barrier with sloping sides
- WKB approximation of a double potential well: turning points
- WKB approximation of double-well potential: wave functions
- WKB approximation of the harmonic oscillator
- WKB approximation and the hydrogen atom
- WKB approximation and the power law potential
- WKB approximation and the radial equation
- WKB approximation and the reflectionless potential
- WKB approximation: tunneling
- WKB approximation: turning points
- WKB approximation at a turning point with decreasing potential
- The wave function as a probability
- The wave function: Born’s conditions
- The wave function: making energy measurements
- Wave-particle duality
- Zeeman effect: degenerate perturbation theory for n = 2
- Zeeman effect for n = 3: general case
- Zeeman effect for n = 3: strong field
- Zeeman effect for n = 3; weak field
- Zeeman effect for l=0
- Zeeman effect: the n=2 line in hydrogen
- Zeeman effect: strength of magnetic field
- Zeeman effect: strong field
- Zeeman effect: weak field

AnonymousI am going through a QM course and not having an easy time of it. Thanks for putting so much effort into this blog, Its much more comprehensive that many.

Cheers from BC

AnonymousI’m new here and I like your stuff. However, some of your blogs are sequential, but the index is alphabetic. It would be nice if under a topic, say harmonic oscillator, the sub-topics were listed sequentially. Or maybe related blogs grouped into chapters.

growescienceThere are already various ways you can get posts related to a given topic. Use the ‘Search’ box in the left panel, or on the index page use your browser’s search option (Control+F on Windows). The problem indexes for the various textbooks I’m working through provide a grouping of posts into chapters. I’ve also tried to tag each post with relevant keywords, so you can do a tag search.

Links to articles referenced in a post are to be found within that post so it shouldn’t be too hard to find a sequence.