PX4087.5 Relativistic Quantum Mechanics
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.
 Introductory Remarks. Revision of relativity, electromagnetism and quantum mechanics; problems with the nonrelativistic Schrödinger equation; unnaturalness of spin in NRQM and the Pauli Hamiltonian; phenomenology of relativistic quantum mechanics, such as pair production
 Klein Gordon Equation. Derivation of the KleinGordon equation; continuity equation and the KleinGordon current; problems with the interpretation of the KleinGordon Equation
 The Dirac Equation. Derivation of the Dirac equation; the quantum phenomenon of spin; gamma matrix algebra and equivalence transformations
 Solutions of the Dirac Equation.
The helicity operator and spin; normalisation of Dirac spinors; Lorentz transformations of Dirac spinors; interpretation of negative energy states  Applications of Relativistic Quantum Mechanics.
The gyromagnetic ratio of the electron; nonrelativistic 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
Indicative reading list
R.Feynman, Quantum Electrodynamics, Perseus Books 1998;
Bethe and Jackiw “Intermediate Quantum Mechanics”, Perseus Books 1997.
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  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 B1
Weighting  Study time  

2 hour online examination (Summer)  100%  
Answer 2 questions from 3

Feedback on assessment
Personal tutor, group feedback
Courses
This module is Optional for:

TMAAG1PE 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 TMAAG1P9 Postgraduate Taught Interdisciplinary Mathematics
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 1 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)
 Year 4 of UPXAF304 Undergraduate Physics (BSc MPhys)
 Year 4 of UPXAF303 Undergraduate Physics (MPhys)
This module is Option list A for:
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 3 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 4 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Option list B for:
 Year 4 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 4 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
This module is Option list C for:

UMAAG105 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

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 3 of G106 Mathematics (MMath) with Study in Europe
 Year 4 of G106 Mathematics (MMath) with Study in Europe