Introductory description

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 shows the need for quantum field theory (QFT). This is where the wavefunctions of matter and light themselves are quantized (made into operators), which automatically builds in the correct fermionic or bosonic statistics of the underlying fields.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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. Fermionic quantum fields. Many-particle states

4. Superconductivity, spin ordering described using Bogoliubov transformations

5. The role of the density matrix in describing open quantum systems and sub-systems

Learning outcomes

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

• Describe the Klein Gordon and Dirac equations, their significance and their transformation properties
• Explain how some physical phenomena including spin, the gyromagnetic ratio of the electron and the fine structure of the hydrogen atom are accounted for within relativistic quantum mechanics
• Define and manipulate quantum fields
• Describe some many-particle states
• Formulate problems in terms of the density matrix

Indicative reading list

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

View reading list on Talis Aspire

Subject specific skills

Knowledge of mathematics and physics. Skills in modelling, reasoning, thinking.

Transferable skills

Analytical, communication, problem-solving, self-study