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Department of Physics

PX453 Advanced Quantum Theory

Lecturer: Tom Blake
Weighting: 15 CATS

The module sets up the relativistic analogues of the Schrödinger equation and introduces quantum field theory. The best equation to describe an electron, due to Dirac, predicts antiparticles, spin and other surprising phenomena. However, Dirac’s equation also showed the need for quantum field theory (QFT). This is where the wavefunctions of matter and light themselves are quantized (made into operators). QFT automatically builds the correct fermionic or bosonic statistics into the description of a many-particle system.

Aims:

This module should start from the premise that quantum mechanics and relativity need to be mutually consistent. The Klein Gordon and Dirac equations should be derived as relativistic generalisations of Schrödinger and Pauli equations. The module should also introduce quantum fields and illustrate how they can describe phenomena in interacting particle systems.

Objectives:

By the end of the module, students should be able to:

  • Describe the Dirac equation, its significance and its transformation properties
  • Explain how some physical phenomena including spin, the gyromagnetic ratio of the electron and the fine structure of the hydrogen atom can be accounted for using relativistic quantum mechanics
  • Understand interactions between electrons in atoms and molecules
  • Be able to work with quantum fields

Syllabus:

  1. Introduction to Relativistic Quantum Mechanics (QM). Problems with the non-relativistic QM; phenomenology of relativistic quantum mechanics, such as pair production. Derivation and interpretation of the Klein- Gordon Equation

  2. The Dirac Equation (DE). Derivation of the DE; spin; gamma matrices and equivalence transformations; Solutions of the DE; Helicity operator and spin; Dirac spinors; Lorentz transformation; interpretation of negative energy states; non-relativistic limit of the Dirac equation; gyromagnetic ratio of electron; fine structure of the hydrogen atom.

  3. Introduction to 2nd Quantisation. Creation and annihilation operators, harmonic oscillator. Spin-statistics theorem. Description of the N-electron state: electronic structure and Bloch’s theorem; electron exchange and correlation, Hartree-Fock; Feynman rules and diagrams.

  4. Landau Fermi Liquid Theory. Notion of the quasiparticle, low temperature properties. Landau Fermi liquid parameters, the Stoner criterion. Application to (some of) normal 3He, magnetism, superconductivity and BCS theory

Commitment: 30 Lectures

Assessment: 2 hour examination

Recommended Text: R.Feynman, Quantum Electrodynamics, Perseus Books 1998
A. Altland & B. D. Simons, Condensed Matter Field Theory, Cambridge University