**Quantum mechanics**

Vector subspace and eigenspace

Completeness of a vector space

Spectral decomposition of an operator

Matrix elements of an operator

Eigenfunctions and eigenvalue equations

Linear combination of wavefunctions

Postulates of quantum mechanics

One-particle, time-independent Schrodinger equation

One-particle, time-independent Schrodinger equation in spherical coordinates

One-particle, time-dependent Schrodinger equation

Relation between magnetic dipole moment and angular momentum

Classical orbital angular momentum

Quantum orbital angular momentum operators (Cartesian coordinates)

Quantum orbital angular momentum operators (spherical coordinates)

Commutation relations of quantum orbital angular momentum operators

Quantum orbital angular momentum ladder operators

Eigenvalues of quantum orbital angular momentum operators

Matrix elements of angular momentum ladder operators

The vector model of quantum angular momentum

Larmor precession of an electron

Gyromagnetic ratio of the electron

Sequential Stern-Gerlach experiments

Composite systems – beyond spin operators

The singlet and triplet states

Einstein-Podolsky-Rosen paradox

Non-relativistic multi-electron Hamiltonian

Degenerate perturbation theory

Degenerate perturbation theory (an example)

Time-dependent perturbation theory

Relativistic one-electron Hamiltonian

Relativistic many-electron Hamiltonian

Constructing atomic terms (uncoupled representation)

Constructing atomic terms (coupled representation)

Reduction of the two-particle problem to one-particle problems

Binomial theorem and binomial series

Generating function for the Legendre polynomials

Recurrence relations of the Legendre polynomials

Rodrigues’ formula of the Legendre polynomials

Orthogonality of the Legendre polynomials

Normalisation constant of the Legendre polynomials

Associated Legendre polynomials

Generating function for the associated Legendre polynomials

Recurrence relations of the associated Legendre polynomials

Orthogonality of the associated Legendre polynomials

Normalisation constant of the associated Legendre polynomials

Orthogonality of the spherical harmonics

Normalisation constant of the spherical harmonics

Rodrigues’ formula for the Laguerre polynomials

Generating function for the Laguerre polynomials

Recurrence relations of the Laguerre polynomials

Orthogonality of the Laguerre polynomials

Associated Laguerre polynomials

Rodrigues’ formula for the associated Laguerre polynomials

Generating function for the associated Laguerre polynomials

Recurrence relations of the associated Laguerre polynomials

Orthogonality of the associated Laguerre polynomials

Normalisation constant of the associated Laguerre polynomials

Total wavefunction of a hydrogenic atom

Degeneracy in hydrogenic atoms