Skip to main content Skip to navigation

PX453-15 Advanced Quantum Theory

Undergraduate Level 4
Module leader
Nicholas d'Ambrumenil
Credit value
Module duration
10 weeks
100% exam
Study location
University of Warwick main campus, Coventry
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

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 the Schrödinger and Pauli equations. The module should introduce quantum fields and illustrate how they can describe phenomena in interacting particle systems, such as superconductivity.

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

Study time

Type Required
Lectures 30 sessions of 1 hour (20%)
Private study 120 hours (80%)
Total 150 hours
Private study description

Working through lecture notes, solving problems, wider reading, discussing with others taking the module, revising for exam, practising on past exam papers


No further costs have been identified for this module.

You must pass all assessment components to pass the module.

Assessment group B
Weighting Study time
Advanced Quantum Theory 100%

Answer 3 questions

  • Students may use a calculator
  • Answerbook Pink (12 page)
Feedback on assessment

Personal tutor, group feedback

Past exam papers for PX453


This module is Optional for:

  • Year 4 of UPXA-F303 Undergraduate Physics (MPhys)

This module is Option list A for:

  • UMAA-G100 Undergraduate Mathematics (BSc)
    • Year 3 of G100 Mathematics
    • Year 3 of G100 Mathematics
    • Year 3 of G100 Mathematics
  • Year 3 of UMAA-G103 Undergraduate Mathematics (MMath)
  • Year 4 of UMAA-G101 Undergraduate Mathematics with Intercalated Year

This module is Option list B for:

  • UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)
    • Year 4 of FG31 Mathematics and Physics (MMathPhys)
    • Year 4 of FG31 Mathematics and Physics (MMathPhys)
  • Year 4 of UPXA-F3FA Undergraduate Physics with Astrophysics (MPhys)

This module is Option list C for:

  • UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
    • Year 4 of G105 Mathematics (MMath) with Intercalated Year
    • Year 5 of G105 Mathematics (MMath) with Intercalated Year
  • UMAA-G103 Undergraduate Mathematics (MMath)
    • Year 3 of G103 Mathematics (MMath)
    • Year 3 of G103 Mathematics (MMath)
    • Year 4 of G103 Mathematics (MMath)
    • Year 4 of G103 Mathematics (MMath)
  • Year 4 of UMAA-G106 Undergraduate Mathematics (MMath) with Study in Europe