# Module Catalogue

Throughout the 2021-22 academic year, we will be prioritising face to face teaching as part of a blended learning approach that builds on the lessons learned over the course of the Coronavirus pandemic. Teaching will vary between online and on-campus delivery through the year, and you should read guidance from the academic department for details of how this will work for a particular module. You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.

# PX408-7.5 Relativistic Quantum Mechanics

Department
Physics
Level
Tom Blake
Credit value
7.5
Module duration
5 weeks
Assessment
100% exam
Study location
University of Warwick main campus, Coventry

##### Introductory description

The module sets up the relativistic analogues of the Schrödinger equation and analyses their consequences. Constructing the equations is not trivial - knowing the form of the ordinary Schrödinger equations turns out not to be much help. The correct equation for the electron, due to Dirac, predicts antiparticles, spin and other surprising phenomena. One is the 'Klein Paradox': When a beam of particles is incident on a high potential barrier, more particles can be 'reflected' than are actually incident on the barrier.

##### 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 are derived as relativistic generalisations of Schrödinger and Pauli equations respectively. The Dirac equation will be analysed in depth and its successes and limitations will be stressed.

##### 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. Introductory Remarks. Revision of relativity, electromagnetism and quantum mechanics; problems with the non-relativistic Schrödinger equation; unnaturalness of spin in NRQM and the Pauli Hamiltonian; phenomenology of relativistic quantum mechanics, such as pair production
2. Klein Gordon Equation. Derivation of the Klein-Gordon equation; continuity equation and the Klein-Gordon current; problems with the interpretation of the Klein-Gordon Equation
3. The Dirac Equation. Derivation of the Dirac equation; the quantum phenomenon of spin; gamma matrix algebra and equivalence transformations
4. Solutions of the Dirac Equation.
The helicity operator and spin; normalisation of Dirac spinors; Lorentz transformations of Dirac spinors; interpretation of negative energy states
5. Applications of Relativistic Quantum Mechanics.
The gyromagnetic ratio of the electron; non-relativistic limit of the Dirac equation; fine structure of the hydrogen atom
##### Learning outcomes

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

• Discuss the general nature of Relativistic Quantum Mechanics
• Solve the Dirac equation in simple cases, and explain its significance and transformation properties.
• Explain how some physical phenomena including spin and the Lamb shift can be accounted for using relativistic quantum mechanics

R.Feynman, Quantum Electrodynamics, Perseus Books 1998;
Bethe and Jackiw “Intermediate Quantum Mechanics”, Perseus Books 1997.

##### 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 15 sessions of 1 hour (20%)
Private study 60 hours (80%)
Total 75 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 B2
Weighting Study time
On-campus Examination 100%

##### Feedback on assessment

Personal tutor, group feedback

## Courses

This module is Optional for:

• TMAA-G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
• Year 1 of TMAA-G1P9 Postgraduate Taught Interdisciplinary Mathematics
• Year 1 of TMAA-G1PD Postgraduate Taught Interdisciplinary Mathematics (Diploma plus MSc)
• Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
• Year 1 of TMAA-G1PC Postgraduate Taught Mathematics (Diploma plus MSc)
• Year 4 of UPXA-F304 Undergraduate Physics (BSc MPhys)
• Year 4 of UPXA-F303 Undergraduate Physics (MPhys)

This module is Option list A for:

• Year 1 of TMAA-G1P0 Postgraduate Taught Mathematics
• Year 3 of UMAA-G100 Undergraduate Mathematics (BSc)
• Year 4 of UMAA-G101 Undergraduate Mathematics with Intercalated Year

This module is Option list B for:

• Year 4 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
• Year 4 of UPXA-FG31 Undergraduate Mathematics and Physics (MMathPhys)

This module is Option list C for:

• UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
• Year 3 of G105 Mathematics (MMath) with Intercalated Year
• Year 5 of G105 Mathematics (MMath) with Intercalated Year