LECTURE NOTES

ADVANCED QUANTUM MECHANICS II

**©D.M. Gingrich**

**University of Alberta
Department of Physics
2004**

**Preface**

These are lecture notes for an advanced graduate course in quantum mechanics. I felt the need to write these notes since no single book currently in print covers the syllabus of the course

This course combines the principles of relativity and quantum theory that are necessary to perform calculations of the electromagnetic scattering of electrons and positrons, as well as, the emission and absorption of photons.

I start by introducing the wave equations for spin-0 and spin-1/2 particles. The basic principles of relativistic quantum mechanics are first introduced for scalar particles where the extra spin-degree of freedom does not obscure the new concepts arising from a relativistic treatment of quantum mechanics. The formalism is then redeveloped for spin-1/2 particles in which a rich set of new concepts are revealed. In each case, I emphasis how the relativistic treatment of quantum mechanics and the spin-1/2 degree of freedom are necessary to describe electromagnetic interactions involving the scattering of electrons. The shortfalls of the wave-equation approach to relativistic quantum mechanics are pointed out and mention is made of how a many-particle quantized field description of the theory is necessary.

The field theoretical approach to quantum mechanics is not investigated in this course but rather a heuristic approach using the propagator formalism developed by Feynman and Stückelburg is used. The Feynman rules for quantum electrodynamics are developed by example. This is an intuitive and practical approach that gets the student doing calculations quickly.

It is my belief that physics at an advanced level is learned by participation and in the case of this theoretical course by performing numerous calculations to develop hands-on experience. Many derivations have been worked out in detail. This will be a benefit to students wanting to study the subject on their own. It is hoped that by studying the material in this course and working through the problems, that the student will gain the necessary background to pursue further study and research in theoretical or particle physics.

Problems at the end of each chapter can be filling in math, proofs of expressions in the main text, proofs of identities, or extensions to ad-hoc models of unusual cases of the theory.

- Contents
- Introduction and Background

- Relativistic Quantum Mechanics
- Klein-Gordon Equation
- Wave Equation for a Spin-0 Particle
- Lorentz Invariance
- Solutions of the Klein-Gordon Equation
- General Solution of the Klein-Gordon Equation
- Conserved Current and Charge
- Orthogonality and Normalization
- Interaction with the Electromagnetic Field
- Klein-Gordon Equation in Schrödinger Form
- Zitterbewegung
- Klein Paradox for Spin-0 Particles
- Coulomb Interaction
- Homogeneous Green Function
- Completeness
- Summary
- Problems

- Dirac Equation
- Wave Equation for a Spin-1/2 Particle
- Current Conservation
- Dirac Particle at Rest
- Electromagnetic Interaction
- Nonrelativistic Limit of the Dirac Equation
- Constants of the Motion
- Free Motion of a Dirac Particle
- Covariant Form of the Dirac Equation
- Proof of Covariance
- Covariance of the Continuity Equation
- Adjoint Spinor
- Bilinear Covariants
- Plane-Wave Solutions
- Projection Operators for Energy and Spin
- Hole Theory
- Charge Conjugation
- Time Reversal
- Combined PCT Symmetry
- Free-Particle Solutions and Wave Packets
- Klein Paradox for Spin-1/2 Particles
- Problems

- Klein-Gordon Equation
- Quantum Electrodynamics
- Propagator Methods
- QED Processes
- Lifetimes and Cross-Sections
- Coulomb Scattering of Electrons
- Coulomb Scattering of Positrons
- Photons
- The Substitution Rules
- Lorentz-Invariant Phase Space
- Electron Scattering from a Dirac Proton (the effect of recoil
- Bremsstrahlung
- Photon-Electron Scattering
- Pair Annihilation into Gamma Rays
- Pair Production
- Electron-Electron Scattering
- Electron-Positron Scattering
- Polarization in Electron Scattering
- Higher-Order Calculations: Tree and Loops
- Feynman Rules for Tree Graphs
- Problems

- Appendices

- Bibliography
- Index
- About this document ...

2004-03-18