FIX University

Part II Principles of Quantum Mechanics

Further information about this course is available on the Department of Mathematics course pages. Examples papers are available on the DAMTP Examples page.

Dirac formalism

Bra and ket notation, operators and observables, probability amplitudes, expectation values, complete commuting sets of operators, unitary operators. Schrödinger equation, wave functions in position and momentum space. [3]

Time evolution operator, Schrödinger and Heisenberg pictures, Heisenberg equations of motion. [2]

Harmonic oscillator

Analysis using annihilation, creation and number operators. Significance for normal modes in physical examples. [2]

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. [3]

Perturbation theory

Time-independent theory; second order without degeneracy, first order with degeneracy. [2]

Angular momentum

Analysis of states |jm〉from commutation relations. Addition of angular momenta, calculation of Clebsch–Gordan coefficients. Spin, Pauli matrices, singlet and triplet combinations for two spin half states. [4]

Translations and rotations

Unitary operators corresponding to spatial translations, momenta as generators, conservation of momentum and translational invariance.  Corresponding discussion for rotations. Reflections, parity, intrinsic parity. [3]

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¤. [3]

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. [2]

Books

† E. Merzbacher Quantum Mechanics, 3rd edition. Wiley 1998 (£36.99 hardback).

† B.H. Bransden and C.J. Joachain Quantum Mechanics, 2nd edition. Pearson (£37.99 paperback).

J. Binney and D. Skinner The Physics of Quantum Mechanics. Cappella Archive, 3rd edition (£21.75)

P.A.M. Dirac The Principles of Quantum Mechanics. Oxford University Press 1967, reprinted 2003 (£24.99 paperback)

C.J. Isham Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press 1995 (£14.00 paperback)