Andreas Brandhuber

This course [official No. 4241 / QMUL No. PHY-414] is taught as part of the University of London Intercollegiate Physics MSci programme.

Teaching Term and Class Location:
Second Term (9 January - 24 March 2006)
Lectures will be held at University College in room UCL Physics A1 on Fridays from 2pm - 5pm. For more details how to get there consult the Intercollegiate Physics MSci webpage.

Aims and Objectives:

Aims:

This course provides a first introduction into the unification of last century's groundshaking revolutions in physics: special relativity and quantum mechanics. Relativistic wave equations for particles of various spins are derived and studied, and the physical interpretation of their solutions are analyzed. Students will learn about propagator theory and relativistic scattering amplitudes in various physically relevant examples. Finally, basic concepts of Quantum Field Theory, which resolve certain puzzles encountered in the study of Relativistic Quantum Mechanics, are introduced.

Objectives:

A student who has satisfactorily completed the course should be able:

to name examples of symmetries and to know how they are related to conserved quantities in quantum mechanics.
to use the four-vector notation and to know the concepts of Lorentz invariance and covariance.
to write down the Klein-Gordon equation and to derive its continuity equation and plane wave solutions.
to write down the Dirac equation, also in its covariant form; to understand the continuity equation, in particular to be aware of the existence of a positive probability density in contrast to the Klein-Gordon equation; to derive plane wave solutions of the Dirac equation and to explain spin; to know about the application of the massless Dirac equation.
to know about discrete symmetries C, P & T and their physical importance.
to explain the meaning of negative energy solutions and the hole theory; to explain Feynman's interpretation of the Klein-Gordon equation.
to master the basic notions of propagator theory and scattering; to understand the definitions of differential cross-sections and transition amplitudes; to follow the derivation of scattering amplitudes in some important examples: electron scattering off a Coulomb potential, electron-proton scattering.
to use Feynman rules to calculate scattering amplitudes for simple processes.
to write down the Lagrangian of a free relativistic scalar particle and to derive the equations of motion and the energy momentum tensor; to quantize the Klein-Gordon field using creation and annihilation operators.

Syllabus:

Quantum Mechanics and Special Relativity (part revision): (~ 6 hours)

Schroedinger equation, wavefunctions, operators/observables, symmetries and conservation laws in QM; short introduction to Special Relativity: 4-vector notation, Lorentz transformations (LT), Lorentz invariance/covariance, LT of the electromagnetic field

Relativistic Wave equations: (~ 10 hours)

Klein-Gordon equation and probability density; Dirac equation, covariance and probability density, non-relativistic limit, spin, Feynman notation, plane wave solutions, Lorentz transformations of plane wave solutions; hole theory and anti-particles, vacuum polarisation; discrete symmetries: C & P & T symmetry and their relevance for electromagnetic and weak interactions, Dirac covariants; wave equations for massless fermions, neutrinos; Feynman interpretation of the Klein-Gordon equation; Dirac equation in an electromagnetic field, magnetic moment of electron, relativistic spectrum of Hydrogen atom.

Relativistic Scattering: (~ 12 hours)

propagator theory, scattering amplitudes, free and interacting electron propagator, scattering amplitudes for electrons and positrons; electron in a Coulomb potential, transition probabilities, differential cross sections, trace theorem, Mott cross-section, Coulomb scattering of a positron; electron scattering from a Dirac proton, proton propagator, electromagnetic transition currents, differential cross-section for electron-proton scattering, other scattering processes; Feynman diagrams

Quantum Field Theory: (~ 5 hours)

Classical field theory, Noether theorem, stress-energy tensor; canonical quantisation of the Klein-Gordon field and the Dirac fermion, creation and annihilation operators, vacuum energy, Casimir energy, spin-statistics connection.

Lecture Notes:

Lecture notes will appear here, updated roughly once a week.
Last year's lecture notes can be found here pdf

Assessment:

Written examination of 2 1/2 hours contributing 90% and coursework contributing 10%

Coursework:

During the semester approximately 8 homework assignments will be given. The best 7 will be used for the assessment

Assignment 1: pdf (issued 12/1/2006, hand in by 27/1/2006) solution
Assignment 2: pdf (issued 19/1/2006, hand in by 3/2/2006) solution
Assignment 3: pdf (issued 30/1/2006, hand in by 10/2/2006) solution
Assignment 4: pdf (issued 6/2/2006, hand in by 17/2/2006) solution
Assignment 5: pdf (issued 13/2/2006, hand in by 24/2/2006) solution
Assignment 6: pdf (issued 20/2/2006, hand in by 3/3/2006) solution
Assignment 7: pdf (issued 27/2/2006, hand in by 10/3/2006) solution
Assignment 8: pdf (issued 6/3/2006, hand in by 17/3/2006) solution

Office Hours: by appointment (email: a.brandhuber@qmul.ac.uk)

My office is in Physics 214 at QMUL.

Exam:

Exams of previous years can be found on the Intercollegiate Physics MSci webpage. The exam paper will be accompanied by a formula sheet.

Recommended Books:

J. Bjorken and S. Drell, "Relativistic quantum mechanics" and "Relativistic quantum fields", McGraw-Hill. (an old but reliable source)

L.H. Ryder, "Quantum Field Theory", Cambridge University Press. (A nice and very pedagocical book, Chapters 1-6 are relevant for our course.)

M.E. Peskin and D.V. Schroeder, "An Introduction to Quantum Field Theory", Addison-Wesley.
(Very good book, which covers much more material than the course. This is THE book you should buy, if you have money for only one book on the subject and you are more serious about theoretical particle physics and/or phenomenology. It is less pedagocial than Ryder, however.)

S. Weinberg, "The Quantum Theory of Fields" in 3 Volumes., Cambridge University Press. (Advanced book on Quantum Field Theory from the Master. Probably overwhelming for a first QFT book. The first chapters of the first volume are relevant for the course.)

C. Itzykson and J.-B. Zuber, "Quantum Field Theory", (Another classic but less pedagogical than Peskin-Schroeder; in particular Sections 1, 2 & 3 have a lot of overlap with the course.)

Relativistic Quantum Mechanics

MSci 4241/QMUL PHY-414 (Winter 2006)