PX45315 Advanced Quantum Theory
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 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.

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

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; nonrelativistic limit of the Dirac equation; gyromagnetic ratio of electron; fine structure of the hydrogen atom

Introduction to 2nd Quantisation. Creation and annihilation operators, harmonic oscillator. Spinstatistics theorem. Fermionic quantum fields. Manyparticle states

Superconductivity, spin ordering described using Bogoliubov transformations

The role of the density matrix in describing open quantum systems and subsystems
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 manyparticle 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, problemsolving, selfstudy
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
Costs
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  Eligible for selfcertification  

Advanced Quantum Theory  100%  No  
Answer 3 questions

Feedback on assessment
Personal tutor, group feedback
Courses
This module is Optional for:
 Year 4 of UPXAF303 Undergraduate Physics (MPhys)
This module is Option list A for:
 Year 3 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 3 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 4 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Option list B for:
 Year 4 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 4 of UPXAF3FA Undergraduate Physics with Astrophysics (MPhys)
This module is Option list C for:

UMAAG105 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

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)
 Year 4 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe