- A few statistics on the first 25 digits of pi
- Adding a constant to the potential introduces a phase factor
- Adding two spin-1/2 systems - product and total-s bases
- Adiabatic approximation: higher order corrections
- Airy functions and the bouncing electron
- Alpha decay using the WKB approximation
- Angular equation - alternative solution
- Angular momentum - commutators
- Angular momentum - commutators with position and momentum
- Angular momentum - eigenfunctions
- Angular momentum - eigenvalues
- Angular momentum - Poisson bracket to commutator
- Angular momentum - raising and lowering operators
- angular momentum and parity
- Angular momentum and torque
- Angular momentum as a generator of rotations
- Angular momentum as an eigenvector problem
- Angular momentum commutators in hydrogen
- angular momentum in 3-d expectation values and uncertainty principle
- Angular momentum in three dimensions
- Angular momentum of circular motion
- angular momentum raising and lowering operators from rectangular coordinates
- 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 of added spins
- Angular momentum: probabilities of eigenvalues in two dimensions
- Angular momentum: restriction to integer values
- Anti-hermitian operators
- Atomic wavefunctions: symmetrization
- Average and standard deviation
- Average rate of change of angular momentum in magnetic field
- Band structure of solids: degeneracy of states
- Band structure of solids: negative energies
- Band structure of solids: numerical solution
- Berry's phase: definition and value for a spin-1 particle in a magnetic field
- Blackbody radiation
- Born approximation for a spherical delta function shell
- Born approximation in one dimension
- Born approximation of delta function well and finite square well
- 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
- Changing the position basis with a unitary transformation
- Clebsch-Gordan coefficients - examples
- Clebsch-Gordan coefficients for addition of spin-1/2 and general L
- Clebsch-Gordan coefficients for higher spin
- Clebsch-gordan coefficients from lowering operators
- Combining translations and rotations
- Commutators: a few theorems
- Complex exponentials and trig functions
- Compound systems of fermions and bosons
- Conditions for a transformation to be canonical
- Continuity of the wave function - Born's conditions revisited
- Continuous probability distribution: needle on a pivot
- Correspondence between classical and quantum transformations
- Coupled masses on springs - a solution using matrix diagonalization
- Coupled masses on springs - properties of the propagator
- Coupling of proton's magnetic moment to external field
- Cyclic coordinates and Poisson brackets
- Decoupling the two-particle Hamiltonian
- Degeneracy pressure in a solid
- Degenerate eigenvalues and Gram-Schmidt orthogonalization
- Degenerate perturbation in 3 state system
- Degenerate perturbation theory: two states
- Degenerate solutions don't exist in one dimension
- Delta function - Fourier transform
- Delta function in time perturbation
- Delta function potential - moving delta function
- Delta function well as limit of finite square well
- Delta function well: bound state - uncertainty principle
- Delta function well: statistics
- Delta-function well - bound state
- Delta-function well - scattering
- Density matrix
- Derivatives
- Determinant and trace of normal operators
- Determinate states
- Diagonalization of matrices
- Differential operator - eigenvalues and eigenstates
- Differential operators - matrix elements and hermiticity
- Dirac delta function
- Dirac delta function - simple examples
- Dirac equation: derivation
- Direct product of two vector spaces
- Direct product of vector spaces: 2-dim examples
- Double delta function well
- Double delta function well - scattering states
- Dysprosium electron configuration
- Earth-Sun system as a quantum atom
- Effective magnetic field in rotating frame - axis of rotation not parallel to field
- Eigenvalues and eigenvectors
- Eigenvalues and eigenvectors - examples
- Eigenvalues and eigenvectors of the 2-d rotation operator
- Eigenvalues of angular momentum
- Eigenvalues of two-dimensional angular momentum
- 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 in a 2-d infinite square well
- Electron gas: a crude model of a solid
- Electron in a precessing magnetic field
- Electron pressure in a neutron star
- Electron pressure in a white dwarf star
- Energy & wave functions - a few theorems
- Energy states: bound and scattering states
- Energy-time uncertainty principle - example
- Energy-time uncertainty principle: Gaussian free particle
- Energy-time uncertainty principle: infinite square well
- Energy-time uncertainty: an alternative definition
- Ensemble of electrons in magnetic field
- Every attractive 1-dimensional potential has a bound state
- Every spin-1/2 spinor is an eigenket of some spin operator
- 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 in the infinite square well
- Fermions and bosons: counting states
- Fermions and bosons: n-particle systems
- Feynman-Hellmann theorem and the harmonic oscillator
- Feynman-Hellmann theorem: hydrogen atom mean values
- Fine structure constant
- Fine structure of a spectral line in hydrogen
- 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
- Finite drop potential
- Finite rotations about an arbitrary axis in three dimensions
- Finite spherical well
- Finite square barrier - scattering
- Finite square well - normalization
- Finite square well - numerical solution
- Finite square well - scattering
- Finite square well: bound states & even wave functions
- Finite square well: bound states & odd wave functions
- Finite square well: bound states & odd wave functions
- Finite step potential - scattering
- Finite transformations: correspondence between classical and quantum
- Finite vector spaces: matrix elements
- 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
- Free particle - travelling wave packet
- Free particle in momentum space
- Free particle in spherical coordinates
- Free particle in spherical coordinates finding the solutions
- Free particle in the position basis
- free particle moving in the z direction
- Free particle propagator from a complete path integral
- Free particle revisited: solution in terms of a propagator
- Free particle: Gaussian wave packet
- Functions of hermitian operators
- Fusion with a muon-deuteron system
- Gaussian distribution
- General 2x2 matrix in terms of pauli matrices
- 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 life of a beer can
- Half-harmonic oscillator
- Hamilton's equations of motion under a regular canonical transformation
- Hamiltonian and observables in three-state system
- Hamiltonian for the electromagnetic force
- Hamiltonian for the two-body problem
- Hamiltonian for three-state system
- Hamiltonian formalism and Legendre transformations
- Hamiltonian in non-rectangular coordinates
- Hamiltonian in two-level system
- Hamiltonian matrix elements
- Hamiltonians for harmonic oscillators
- Harmonic oscillator - asymptotic solution
- Harmonic oscillator - change in spring constant
- Harmonic oscillator - classical physics
- Harmonic oscillator - eigenfunctions in momentum space
- Harmonic oscillator - example starting state
- Harmonic oscillator - Hermite polynomials
- Harmonic oscillator - mean position and momentum
- Harmonic oscillator - mixed initial state
- Harmonic oscillator - mixed initial state and Ehrenfest's theorem
- Harmonic oscillator - position, momentum and energy
- Harmonic oscillator - probability of being outside classical region
- Harmonic oscillator - raising and lowering operator calculations
- Harmonic oscillator - raising and lowering operators as functions of time
- Harmonic oscillator - series solution
- Harmonic oscillator - series solution revisited
- Harmonic oscillator - summary
- Harmonic oscillator - three lowest stationary states
- Harmonic oscillator - zero-point energy from uncertainty principle
- Harmonic oscillator energies and eigenfunctions derived from the propagator
- Harmonic oscillator excited states - numerical solution
- Harmonic oscillator ground state - numerical solution
- Harmonic oscillator in 2 dimensions: comparison with rectangular coordinates
- Harmonic oscillator in 2-d and 3-d, and in polar and spherical coordinates
- Harmonic oscillator in 3-d - rectangular coordinates
- Harmonic oscillator in 3-d: spherical coordinates
- Harmonic oscillator in a magnetic field
- Harmonic oscillator in an electric field
- Harmonic oscillator: algebraic normalization of raising and lowering operators
- Harmonic oscillator: algebraic solution
- Harmonic oscillator: coherent states
- Harmonic oscillator: first order perturbation
- Harmonic oscillator: Hermite polynomials and orthogonality of eigenfunctions
- Harmonic oscillator: matrix elements
- Harmonic oscillator: matrix elements using Hermite polynomials
- Harmonic oscillator: mixture of two lowest states
- Harmonic oscillator: momentum space functions and Hermite polynomial recursion relations from raising and lowering operators
- Harmonic oscillator: relativistic correction
- Harmonic oscillator: Schrödinger's exact solution
- Harmonic oscillator: statistics
- Helium atom
- Helium atom using the variational principle
- Helium atom: electron-electron interaction
- Helium atom: parahelium and orthohelium
- Hermite polynomials - generation
- Hermite polynomials - recursion relations
- Hermite polynomials – the Rodrigues formula
- Hermitian conjugate of an operator
- Hermitian matrices - example with 4 matrices
- Hermitian operators
- Hermitian operators - a few examples
- Hermitian operators - a few theorems
- Hermitian operators - a few theorems
- Hermitian operators - equivalence of conditions
- Hermitian operators: common eigenfunctions implies they commute
- Hermitian operators: periodic function
- Hilbert space - power functions
- Hund's rules
- Hybrid infinite-finite square well
- Hydrogen atom - complete wave function
- 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 - radial equation
- Hydrogen atom - radial function examples
- Hydrogen atom - series solution and Bohr energy levels
- Hydrogen atom - spectrum
- Hydrogen atom - wave function examples
- hydrogen atom a sample wave function
- Hydrogen atom in a crystal lattice
- hydrogen atom radial function at large r
- Hydrogen atom: coincident spectral lines
- Hydrogen atom: combined position and spin state
- Hydrogen atom: powers of the momentum operator
- Hydrogen atom: probability of finding electron inside the nucleus
- Hydrogen atom: radial functions for large l
- Hydrogen atom: wave function example 3
- Hydrogen molecule ion
- Hydrogen molecule ion - different trial function
- Hydrogen molecule ion – oscillation of the protons
- Hydrogen-like atoms
- Hyperfine interaction in hydrogen - a rough calculation
- Hyperfine splitting and the 21 cm line of hydrogen
- Hyperfine splitting in deuterium
- Identical particles - bosons and fermions revisited
- Identical particles: fermions and bosons
- Impulse approximation in scattering theory
- Infinite spherical well - numerical solutions
- Infinite spherical well - spherical Bessel functions
- 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 - expanding well
- Infinite square well - force to decrease well width
- Infinite square well - minimum energy
- Infinite square well - numerical solution
- Infinite square well - particle in left half
- Infinite square well - phase difference
- Infinite square well - triangular initial state
- Infinite square well - uncertainty principle
- Infinite square well in three dimensions
- Infinite square well with delta function barrier
- Infinite square well with triangular initial state using delta function
- Infinite square well with variable delta function barrier: ground state energy
- Infinite square well with variable delta function barrier: location of the particle
- Infinite square well: 2 particle systems
- Infinite square well: momentum
- Infinite square well: momentum space wave functions
- Infinitesimal rotations in canonical and noncanonical transformations
- Inner products and Hilbert spaces
- Integral form of the Schrödinger equation
- Integral form of the Schrödinger equation: ground state of hydrogen
- Integral of inner product of two wave functions is constant in time
- Integrals
- Invariance of Euler-Lagrange and Hamilton's equations under canonical transformations
- Invariance of symmetric and antisymmetric states; exchange operators
- Inverses of linear operators
- isotropic harmonic oscillator in 3 d use of spherical harmonics
- kinematics of spin hilbert space for an electron
- Kinetic energy
- Kramers's relation for averages of radial powers in hydrogen
- Kramers's relation: application to hydrogen mean values
- Lagrangian for a spherically symmetric potential energy function
- Lagrangian for the two-body problem
- Lagrangians for harmonic oscillators
- Legendre polynomials - orthogonality
- Principle of least action - where does it come from?
- Legendre polynomials: generation by Gram-Schmidt process
- Levi-Civita antisymmetric tensor, vector products and systems of 3 fermions
- Limits
- linear combinations of spherical harmonics probabilities
- Linear functionals and adjoint operators
- Linear operators
- Linear operators: null space, range, injectivity and surjectivity
- Magnetic moment in oscillating magnetic field
- Magnetic resonance
- Matrix elements: example
- Matrix representation of linear operators; matrix multiplication
- Matrix representation of linear operators: change of basis
- Momentum space in 3-d
- Momentum space representation of finite wave function
- Momentum space: another example
- Momentum space: harmonic oscillator
- Momentum space: mean position
- Momentum: eigenvalues and normalization
- Non-denumerable basis: position and momentum states
- nondegenerate states in 3-d spherically symmetric systems
- Normal operators
- Optical theorem
- Orthonormal basis and orthogonal complement
- Parity transformations
- Partial waves in three dimensions: hard sphere scattering
- Particle on a circular wire
- Passive, regular and active transformations. Invariance of the Hamiltonian and generators of transformations
- Path integral formulation of quantum mechanics: free particle propagator
- Path integral to Schrödinger equation for a vector potential
- Path integrals for special potentials; use of classical action
- Pauli matrices: A useful identity
- Pauli matrices: commutation and anticommutation properties
- Pauli matrices: examples of linear combinations
- Pauli matrices: properties
- Periodic potentials: Bloch's theorem and the band structure of solids
- Periodic table
- Perturbation due to finite size of the proton in hydrogen
- Perturbation of 3-d square well
- Perturbation theory and the variational principle
- Perturbation theory for higher-level degenerate systems
- Perturbing a particle on a circular wire
- Perturbing the 3-d harmonic oscillator
- 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
- Poisson brackets are invariant under a canonical transformation
- Poisson brackets to commutators: classical to quantum
- Position and momentum
- Position and momentum unit operators
- Position operator - eigenfunctions
- Postulates of quantum mechanics: momentum
- Postulates of quantum mechanics: Schrödinger equation and propagators
- Postulates of quantum mechanics: states and measurements
- Potential energy
- Potential versus potential energy
- Probability current
- Probability current in 3-d
- Probability current with complex potential
- Probability current: a few examples
- Projection operators
- Projection operators
- Projection operators for general L + spin-1/2
- Projection operators for spin-1/2 + spin-1/2
- Propagator for a Gaussian wave packet for the free particle
- Quantum dots
- Quantum revival time
- Quantum scattering: partial wave analysis
- Quantum scattering: scattering amplitude and differential cross section
- Quantum versus classical mechanics in solids and gases
- radial function for large r
- radial function for small r
- Radially symmetric potentials, angular momentum and centrifugal force
- Reflectionless potential
- Relation between action and energy
- Rigid rotor in quantum mechanics
- Rotation matrices - matrix elements
- Rotation of a vector wave function
- Rotation of spinor about arbitrary direction
- Rotational invariance in two dimensions
- Rotational transformations using passive transformations
- rotations in 3-d classical and quantum rotations compared
- rotations in 3-d euler angles
- Rotations through a finite angle; use of polar coordinates
- Rubber band helium
- runge lenz vector and closed orbits
- Rutherford scattering
- Scattering from the Yukawa potential
- Scattering matrix
- Schrödinger equation - a few theorems
- Schrödinger equation - minimum energy
- Schrödinger equation for 2 particles - separation of variables
- Schrödinger equation in three dimensions - spherical harmonics
- Schrödinger equation in three dimensions - the radial equation
- Schrödinger equation: the motivation
- Second order Born approximation in scattering theory
- Second order non-degenerate perturbation theory
- Second-order correction to zeeman effect in hydrogen
- 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
- sizes of elementary particles
- Spectral decomposition of operators
- Spectral theorem for normal operators
- spherical bessel functions behaviour for small arguments
- Spherical harmonic at the top of the ladder
- Spherical harmonic using the raising operator
- Spherical harmonics - examples
- Spherical harmonics - more examples
- spherical harmonics from power series examples for m0
- spherical harmonics rotation about the x axis
- Spherical harmonics using the lowering operator
- Spherical harmonics: normalization
- Spherically symmetric potentials a simple example
- Spherically symmetric potentials hermiticity of the radial function
- Spherical tensor operators; commutators
- Spherical tensor operators; a scalar operator
- Spin - expectation values of components
- Spin - introduction
- Spin - statistical calculations
- Spin 1
- Spin 1/2 along an arbitrary direction
- Spin 1/2 particle in a magnetic field
- Spin 1/2 particle in time-varying magnetic field
- Spin 1/2: minimum uncertainty
- Spin 1/2: spin components
- Spin ½
- Spin 3/2
- Spin and quarks
- Spin flip of electron in magnetic field
- Spin matrices: general case
- Spin: the x and y components
- Spinor in oscillating magnetic field - part 1
- Spinor in oscillating magnetic field - part 2
- Spontaneous emission from n=3 to n=1 in hydrogen
- Spontaneous emission from the zero point field
- Spontaneous emission rates for hydrogen: general solution
- Spontaneous emission rates for the hydrogen atom
- Spontaneous emission: Einstein's argument
- 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 boson states
- Statistical mechanics in quantum theory: counting 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
- Stern-gerlach experiment
- Stimulated emission of radiation at high frequencies
- Stimulated emission of radiation: lasers
- Subspaces and direct sums
- Symmetry of states formed from two equal spins
- The adiabatic approximation in quantum mechanics
- The classical limit of quantum mechanics; Ehrenfest's theorem
- The energy-time uncertainty relation
- The free particle
- The free particle as a wave packet
- The free particle: probability current
- The infinite square well (particle in a box)
- The need for quantum theory
- The path integral is equivalent to the Schrödinger equation
- The Schrödinger equation
- The time-independent Schrödinger equation
- The uncertainty principle
- The variational principle in quantum mechanics
- The wave function as a probability
- Thermodynamics of harmonic oscillators - classical and quantum
- Time dependent Schrödinger equation: two-state systems
- Time reversal, antiunitary operators and Wigner's theorem
- Time translation and conservation of energy
- Time-dependent perturbation of the infinite square well
- Time-dependent perturbation theory for a multi-level system
- Time-dependent perturbation theory: general two-state solution
- Time-dependent perturbation theory: iterative solution
- Time-dependent perturbation theory: switching a perturbation on and off
- Time-dependent perturbation theory: transition probabilities in a multi-state system
- Time-dependent perturbation uniform in space
- Time-dependent propagators
- Time-dependent Schrödinger equation: switching a perturbation on and off
- Total angular momentum - matrix elements and commutation relations
- total angular momentum finite rotations
- total angular momentum is hermitian
- Total-s matrix and eigenstates in product basis
- Transfer matrix
- Translation invariance in two dimensions
- Translation operator from passive transformations
- Translational invariance and conservation of momentum
- Translational invariance in quantum mechanics
- Translations in space and time
- Triangle inequality as an equality
- Triangular wave function: probabilities
- Two-dimensional harmonic oscillator - Part 1
- Two-dimensional harmonic oscillator – Part 2: Series solution
- Uncertainties in the harmonic oscillator and hydrogen atom
- Uncertainty principle - a stronger form
- Uncertainty principle - examples
- Uncertainty principle and an estimate of the ground state energy of hydrogen
- Uncertainty principle in three dimensions
- Uncertainty principle: an example
- Uncertainty principle: condition for minimum uncertainty
- Uncertainty principle: rates of change of operators
- Unitary matrices - some examples
- Unitary operators
- Unitary operators: active and passive transformations of an operator
- Unstable particles: a crude model
- Van der Waals interaction
- Variational principle and the delta function well
- Variational principle and harmonic oscillator: a more general trial function
- Variational principle and the harmonic oscillator
- Variational principle and the harmonic oscillator (quartic wave function)
- Variational principle and the infinite square well
- 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 the hydrogen atom
- Variational principle and l=1 states of the hydrogen atom
- Variational principle and the hydrogen ion: two parameters
- Variational principle and the Yukawa potential
- Variational principle with a two-state hamiltonian
- Vector operators; transformation under rotation
- Vector spaces - number of dimensions
- Vector spaces & linear independence - some examples
- Vector spaces and Hilbert space
- Vector spaces: definitions and examples
- Vector spaces: span, linear independence and basis
- Vibrating string - normal mode analysis
- Vibrational states in a diatomic molecule (HCl)
- Virial theorem
- Virial theorem in 3-d
- Virial theorem in classical mechanics; application to harmonic oscillator
- Wave function: Born's conditions
- Why physics needs calculus
- Wigner-Eckart Theorem
- Wigner-Eckart Theorem - adding orbital and spin angular momenta
- Wigner-Eckart Theorem - examples
- WKB approximation
- WKB approximation - alternative derivation
- WKB approximation - analysis of the overlap region near a turning point
- 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 at a turning point with decreasing potential
- 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: tunneling
- WKB approximation: turning points
- Zeeman effect for l=0
- Zeeman effect for n = 3; weak field
- Zeeman effect for n = 3: general case
- Zeeman effect for n = 3: strong field
- Zeeman effect: degenerate perturbation theory for n = 2
- Zeeman effect: strength of magnetic field
- Zeeman effect: strong field
- Zeeman effect: the n=2 line in hydrogen
- Zeeman effect: weak field