# PX408 Relativistic Quantum Mechanics

## Lecturer: Tim Gershon

## Weighting: 7.5 CATS

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.

**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 Schrödinger and Pauli equations respectively. The Dirac equation should be analysed in depth and its successes and limitations will be stressed.

**Objectives:**

At the end of this module you should:

- have an appreciation of the general nature of Relativistic Quantum Mechanics.
- have an understanding of the Dirac equation, its significance and its transformation properties
- be able to explain how some physical phenomena including spin, the gyromagnetic ratio of the electron and the fine structure of the hydrogen atom can be accounted for using relativistic quantum mechanics

**Syllabus:**

- 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

- 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

- The Dirac Equation
- Deriviation of the Dirac equation; the unavoidable emergence of the quantum phenomena 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; non-relativistic limit of the Dirac equation; fine structure of the hydrogen atom

**Commitment:** 15 Lectures

**Assessment:** 1.5 hour examination

The module has a website.

**Recommended Text:** The course closely follows

R.Feynman, Quantum Electrodynamics, Perseus Books 1998

**Leads from:** PX109 Relativity; PX262 Quantum Mechanics and its Applications

**Leads to:** PX430 Gauge Theories of Particle Physics;