Prerequisites

This course is recommended only for students who have achieved a strong performance in Mathematics as well as Physics in Part IB, or an equivalent qualification. You should be VERY comfortable with the material from the Advanced Quantum Physics course. Some familiarity with Lagrangian mechanics (as discussed in TP1) is also useful but not essential.

Learning Outcomes and Assessment

Working knowledge on Quantum Dynamics, Scattering Theory, Second Quantisation, as well as Density Matrices.

Synopsis

The development of quantum theory during the 20th century led to the introduction of completely new concepts to physics. At the same time, physicists were forced -- sometimes unwillingly -- to adopt myriad new techniques and mathematical ideas. In this course, we'll survey some of these more advanced topics.

Quantum Dynamics
Schrödinger, Heisenberg, interaction picture. The evolution operator and time ordering.
Driven oscillator. Coherent states. A spin-1/2 in a field. Rabi oscillations.
The adiabatic approximation. Landau-Zener transitions. Berry's phase.

Introduction to path integrals
The propagator and the Green's function: free particle and harmonic oscillator.
The method of stationary phase, the JWKB method and the semiclassical limit.

Scattering Theory
Scattering in one dimension. Scattering amplitude and cross section.
Optical theorem. Lippmann-Schwinger equation. Born series.
Partial wave analysis. Bound states.

Identical Particles in Quantum Mechanics
Second quantisation for bosons and fermions.
Single-particle density matrix and density-density correlation function.
Bose-Hubbard model. Bogoliubov transformation.
Interference of condensates.

Density Matrices
Density matrix and its properties. Applications in statistical mechanics.
Density operator for subsystems and entanglement. Quantum damping.

Lie Groups
Symmetries are groups. Lie algebra of generators. Rotations as Lie group
Representations of SO(3), SU(2),Lorentz group SO(1,3) and SL(2,C).

Relativistic Quantum Physics
Klein-Gordon equation. Antiparticles.
Spinors and the Dirac equation. Relativistic covariance.

[Items above in italics are non-examinable content.]

BOOKS

Modern Quantum Mechanics, Sakurai J J (2nd edn Addison-Wesley 1994)

(Advanced) Quantum Mechanics, Schwabl F (4th edn Springer 2007/2008)

Principles of Quantum Mechanics, Shanker R (2nd edn Springer 1994)

Lectures on Quantum Mechanics, Baym G (Benjamin WA 1969)

For mathematical background we’d heartily recommend Mike Stone and Paul Goldbart’s Mathematics for Physicists: A guided tour for graduate students (SUP, 2009).  This contains a lot of advanced material as well as much of what you covered in IB Mathematics.

A great resource for just about anything you may need to know about any of the functions we meet is the NIST Digital Library of Mathematical Functions at http://dlmf.nist.gov.