- 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 an eigenvector problem
- 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
- Canonical transformations in 2-d: rotations and polar coordinates
- Canonical transformations: a few more examples
- Clebsch-Gordan coefficients for higher spin
- Commutators: a few theorems
- Complex exponentials and trig functions
- Conditions for a transformation to be canonical
- Continuity of the wave function – Born’s conditions revisited
- Continuous probability distribution: needle on a pivot
- Coupled masses on springs – a solution using matrix diagonalization
- Coupled masses on springs – properties of the propagator
- Cyclic coordinates and Poisson brackets
- 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
- Determinant and trace of normal operators
- Determinate states
- Diagonalization of matrices
- Differential operators – matrix elements and hermiticity
- Differential operator – eigenvalues and eigenstates
- 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
- Eigenvalues and eigenvectors
- Eigenvalues and eigenvectors – examples
- Eigenvalues and eigenvectors of the 2-d rotation operator
- Electrodynamics in quantum mechanics: gauge transformations
- Electromagnetic force law in quantum mechanics
- Electromagnetic Lagrangian
- 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
- Every attractive 1-dimensional potential has a bound state
- Exchange force: harmonic oscillator
- Exchange force: infinite square well
- Exponentials of operators – Baker-Campbell-Hausdorff formula
- Exponentials of operators – Hadamard’s lemma
- 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
- Functions of hermitian operators
- Fusion with a muon-deuteron system
- Gaussian distribution
- Geometric phase is always zero for real wave functions
- Gram-Schmidt orthogonalization – a couple of examples
- 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
- Hamiltonian formalism and Legendre transformations
- Hamiltonian for the electromagnetic force
- Hamiltonians for harmonic oscillators
- Hamiltonian for the two-body problem
- 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 matrices – example with 4 matrices
- Hermitian operators – a few examples
- Hermitian operators – a few theorems
- Hermitian operators – a few more 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 products and Hilbert spaces
- 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
- Invariance of Euler-Lagrange and Hamilton’s equations under canonical transformations
- Inverses of linear operators
- Kramers’s relation for averages of radial powers in hydrogen
- Kramers’s relation: application to hydrogen mean values
- Lagrangians for harmonic oscillators
- Lagrangian for a spherically symmetric potential energy function
- Lagrangian for the two-body problem
- Laguerre polynomials – normalization
- Legendre polynomials: generation by Gram-Schmidt process
- Linear functionals and adjoint operators
- Linear operators & commutators
- Linear operators: null space, range, injectivity and surjectivity
- Magnetic resonance
- Matrix elements: example
- Matrix representation of linear operators; matrix multiplication
- Matrix representation of linear operators: change of basis
- 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
- Non-denumerable basis: position and momentum states
- Normal operators
- Optical theorem
- Orthonormal basis and orthogonal complement
- 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 and momentum unit operators
- Position operator: eigenfunctions
- Probability current
- Probability current in 3-d
- Projection operators
- Projection operators (linear algebra)
- 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
- Rotation matrices – matrix elements
- 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
- Simultaneous diagonalization of hermitian matrices
- Sinusoidal perturbations in time
- Spectral decomposition of operators
- Spectral theorem for normal 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
- Subspaces and direct sums
- 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
- Triangle inequality as an equality
- Triangular wave function: probabilities
- Uncertainty principle; position-momentum commutator
- 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
- Unitary matrices – some examples
- Unitary operators
- Unitary operators: active and passive transformations of an operator
- 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: definitions and examples
- Vector spaces – number of dimensions
- Vector spaces: span, linear independence and basis
- Vector spaces & linear independence – some examples
- Vector spaces and Hilbert space
- Vibrating string – normal mode analysis
- 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.