# B7.3 Further Quantum Theory - Material for the year 2019-2020

## Primary tabs

Part A Quantum Theory.

16 lectures

### Assessment type:

- Written Examination

This course builds on Part A Quantum Mechanics. The mathematical foundations of quantum theory are developed more deeply than in the Part A course, and general principles regarding the consequences of symmetry for quantum mechanical systems and the nature of identical particles is emphasized. Along the way, simple-but-relevant concepts from the theories of Lie groups, representation theory, and functional analysis are introduced.

In applications to semi-realistic systems, an exact solution is rarely forthcoming, so a number of important approximation techniques are also developed. These are employed to address problems such as the determination of energy levels of the Helium atom and elementary scattering theory.

Student will be able to demonstrate a thorough knowledge and understanding of the mathematical foundations of quantum mechanics, including the consequences of symmetries for the structure of a quantum mechanical system. The will know how to apply this formalism to the analysis of systems of several elementary particles, bound states of several particles, and simple scattering processes. They will be familiar with a variety of approximation methods that can be employed to study quantum mechanical systems where an exact analysis cannot be carried out.

Review of the abstract formulation of quantum mechanics in terms of linear operators on Hilbert spaces.

Systems of several particles: distinguishable and indistinguishable particles; Fermi-Dirac and Bose-Einstein statistics; Pauli exclusion principle.

Symmetries in quantum mechanics: rotations, angular momentum, and spin; addition of angular momentum; Spin-statistics theorem.

Approximation methods: Rayleigh-Schrödinger perturbation theory; variational methods; WKB approximation. Applications of all such methods to simple examples.

Heisenberg, Schrödinger, and interaction/Dyson pictures of time-dependence in quantum mechanics. Time-dependent perturbation theory and the Feynman-Dyson expansion.

Elementary scattering theory.

Keith Hannabuss, *An Introduction to Quantum Theory* (OUP, 1997). Chapter 6.1-6.4, 9.1, 9.3, 9.9, 8.1-8.4, 11.1-11.5, 12.1-12.4. 14.1-14.4, 15.1-15.3, 16.1-16.4.

Steven Weinberg, *Lectures on Quantum Mechanics* (CUP, 2015). Chapter 3.1-3.6, 4.1-4.6, 5.1-5.7, 6.1-6.5, 7.1-7.4.

J. J. Sakurai and Jim Napolitano, *Modern Quantum Mehanics* (CUP, 2017). Chapters 3-7.

David Griffiths and Darrell Schroeter, *Introduction to Quantum Mechanics* (CUP, 2017).

B. Hall, *Quantum Theory for Mathematicians* (Springer, 2013).

Eugene Merzebcher, *Quantum Mechanics* (Wiley, 1970).