# Part II Principles of Quantum Mechanics

## Principles of Quantum Mechanics

### Michaelmas Term, 24 Lectures – Dr E. Pajer

Further information about this course is available on the Department of Mathematics course pages.

Examples papers are available on the DAMTP Examples page.

1. Dirac formalism: Bra and ket notation, operators and observables, probability amplitudes, expectation values, complete commuting sets of operators, unitary operators. Schrodinger equation, wave functions in position and momentum space. 

1. Time evolution operator: Schrodinger & Heisenberg pictures, Heisenberg equations of motion. 

1. Harmonic oscillator: Analysis using annihilation, creation and number operators. Significance for normal modes in physical examples. 

1. Multiparticle systems: Composite systems and tensor products, wave functions for multiparticle systems. Symmetry or antisymmetry of states for identical particles, Bose and Fermi statistics, Pauli exclusion principle. 

1. Perturbation theory: Time-independent theory; second order without degeneracy, first order with degeneracy. 

1. Angular momentum: Analysis of states ljm> from commutation relations. Addition of angular momenta, calculation of Clebsch-Gordan coefficients. Spin, Pauli matrices, singlet and triplet combinations for two spin half states. 

1. Translations and rotations: Unitary operators corresponding to spatial translations, momenta as generators, conservation of momentum and translational invariance. Corresponding discussion for rotations. Reactions, parity, intrinsic parity. 

1. Time-dependent perturbation theory: Interaction picture. First-order transition probability, the golden rule for transition rates. Application to atomic transitions, selection rules based on angular momentum and parity, absorption, stimulated and spontaneous emission of photons. 

1. Quantum basics: Quantum data, qubits, no cloning theorem. Entanglement, pure and mixed states, density matrix. Classical determinism versus quantum probability, Bell inequality for singlet two-electron state, GHZ state.