Change in Syllabus 2004/5

• Introduction/Revision: Mathematical foundations of non-relativistic quantum mechanics. Vector spaces. Operator methods for discrete and continuous eigenspectra. Generalized form of the uncertainty principle. Dirac delta function and delta-function potential.
• Quantum Dynamics: Time development operator. Schrodinger, Heisenberg and interaction pictures. Canonical quantisation and constants of motion. Coordinate and momentum representations. Free particle and simple harmonic oscillator propagators. Introduction to path integral formulation.
• Approximate Methods: Variational methods and their application to problems of interest. The JWKB method and connection formulae, with applications to bound states and barrier penetration. The anharmonic oscillator. Asymptotic expansions.
• Scattering Theory: Scattering amplitudes and differential cross section. Partial wave analysis and the optical theorem. Green functions, weak scattering and the Born approximation. Relation between Born approximation and partial wave expansions. Beyond the Born approximation.
• Density Matrices: Pure and mixed states. The density operator and its properties. Position and momentum representation of the density operator. Spin density matrix and polarisation. Density matrix for the harmonic oscillator. Applications in statistical mechanics.
• Lie Groups: Rotation group, SO(3) and SU(2). SO(4) and the hydrogen atom. Applications to atomic structure. SU(3) and quarks.

• Operator Methods in Quantum Mechanics (2 lectures) Mathematical foundations of non-relativistic quantum mechanics; operator methods for discrete and continuous eigenspectra; generalized form of the uncertainty principle; simple harmonic oscillator; delta-function potential; introduction to second quantization.
• Angular Momentum (2 lectures) Eigenvalues/eigenvectors of the angular momentum operators (orbital/spin); spherical harmonics and their applications; Pauli matrices and spinors; addition of angular momenta.
• Approximation Methods for Bound States (2 lectures) Variational methods and their application to problems of interest; perturbation theory (time-independent and time dependent) including degenerate and non-degenerate cases; *the JWKB method and its application to barrier penetration and radioactive decay.
• Scattering Theory (2 lectures) Scattering amplitudes and differential cross-section; partial wave analysis; the optical theorem; Green functions; weak scattering and the Born approximation; *relation between Born approximation and partial wave expansions; *beyond the Born approximation.
• Identical Particles in Quantum Mechanics (2 lectures) Wave functions for non-interacting systems; symmetry of many-particle wave functions; the Pauli exclusion principle; fermions and bosons; exchange forces; *the hydrogen molecule; scattering of identical particles; *second quantization method for many-particle systems; *pair correlation functions for bosons and fermions; *the self-consistent field method in electronic structure theory.
• Density Matrices (2 lectures) Pure and mixed states; the density operator and its properties; *position and momentum representation of the density operator; *density matrix for the harmonic oscillator; applications in statistical mechanics.

• (1) Removal of chapters on Angular Momentum and Identical Particles
• (2) Inclusion of chapters on Quantum Dynamics and Continuous Groups.