Description

Starting with the second-quantisation formalism, the module uses sophisticated methods (Green functions, Feynman diagrams, and relativistic and non-relativistic quantum field-theories) to analyse the various phaenomonena that arise from the presence of interactions in many-body quantum systems of bosons and fermions, including the Hartree-Fock approximation, the microscopic Bogoliubov theory of superfluidity, spontaneous symmetry-breaking and the BCS theory of superconductivity.

Module Aims

The aim of the module is to introduce the foundations of many-body quantum theory, from both the technical and physical points of view. Although many of the examples are drawn from condensed matter physics, the analogies between these and the theories of high-energy physics will also be emphasised and illustrated.

Intended Learning Outcomes (ILOs)

A student who has passed this module should be able to:

• Module Specific Skills and Knowledge:
  1. quantise fields both on a basis and in a continuum;
  2. describe both fields and particles in a consistent occupation number representation;
  3. use field operators in simple examples;
  4. explain the failings of Hartree-Fock theory and the role played by correlation;
  5. derive and solve the simple Bogluibov condensate equations on the basis of a macroscopically occupied state;
  6. apply quantum field theory techniques to the many-body problem
  7. discuss and explain the physical consequences of the presence of interactions in correlated systems at low temperatures;
• Discipline Specific Skills and Knowledge:
  8. use second-quantisation as a tool for solving quantum mechanical problems;
  9. discuss physical systems within the framework of various quantum mechanical representations;
• Personal and Key Transferable / Employment Skills and Knowledge:
  10. give qualitative descriptions of complicated theories and systems;
  11. develop self-study skills;
  12. use mathematical methods to solve problems.

Syllabus Plan

1. Introduction to Second Quantisation
   1. The quantum harmonic oscillator
   2. Second quantisation of the electromagnetic field: photons
2. Quantum Field Theory of Interacting Bosons
   1. Introduction to the quantum field theory formalism for bosons
   2. Quasiparticles in a system of interacting bosons
   3. Bogoliubov microscopic theory of superfluidity
   4. Theory of the condensed states: Gross-Pitaevski equation
3. Quantum Field theory of Interacting Fermions
   1. Introduction to the quantum field theory formalism for fermions
   2. Quasiparticles in a system of interacting bosons: Hartree-Fock approximation
   3. Cooper instability for electrons with attractive interactions
   4. BCS theory of superconductivity
4. Introduction to Feynman Diagrams
   1. Introduction to single-particle Green's functions at zero temperature
   2. The Feynman-Dyson perturbation theory
   3. Hartree-Fock revisited: diagrammatic approach

Learning and Teaching

Learning Activities and Teaching Methods

Assessment

Resources

The following list is offered as an indication of the type & level of information that students are expected to consult. Further guidance will be provided by the Module Instructor(s).

Core text:

• Not applicable

Supplementary texts:

• Abrikosov A.A. (1975), Methods of Quantum Field Theory in Statistical Physics, Dover, ISBN 978-0-486-63228-5
• Baym G. (1969), Lectures on Quantum Mechanics, Benjamin/Cummings, ISBN 8-053-0664-1
• Bethe H.A. (1986), Intermediate Quantum Mechanics (3^rd edition), Addison-Wesley, ISBN 0-8053-0757-5
• Davydov A.S. (1965), Quantum Mechanics, Pergamon Press, ISBN 978-0-080-13143-6
• Doniach S. and Sondheimer E.H. (1974), Green's Funtion for Solid State Physic, Benjamin, ISBN 8-0532394-5
• Fetter A.L. and Walecka J.D. (2003), Quantum Theory of Many-Particle Systems, Dover, ISBN 978-0-486-42827-7
• Feynman R.P., Leighton R.B. and Sands M. (1965), Lectures on Physics, Vol. III,
• Heitler W. (1954), Quantum Theory of Radiation, Clarendon Press
• Inkson J.C. (1984), Many Body Theory of Solids, Plenum, ISBN 0-306-41326-4
• Pitaevskii L.P. and Lifshitz E.M. (1980), Statistical Physics (Part 2), Butterworth-Heinemann, ISBN 978-0-750-62636-1
• Messiah A. (1981), Quantum Mechanics, Vol. I (12^th edition), North Holland, ISBN 978-0-720-40044-1
• Messiah A. (1981), Quantum Mechanics, Vol. II (1^st edition), North Holland, ISBN 978-0-720-40045-8
• Nozieres P. and Pines D. (1999), Theory of Quantum Liquids, Westview Press, ISBN 978-0-738-20229-7
• Pethick C.J. and Smith H. (2008), Bose-Einstein Condensation in Dilute Gases (2^nd edition), Cambridge University Press, ISBN 978-0-521-84651-6
• Sakurai J.J. and Napolitano J.J. (2010), Modern Quantum Mechanics (2^nd edition), , ISBN 978-0-805-38291-4
• Schrieffer J.R. (1971), Theory of Superconductivity (3^rd edition), Westview Press, ISBN 978-0-7-3820120-7

ELE:

• http://newton.ex.ac.uk/redirects/ELE/phym013

Further Information

Prior Knowledge Requirements

Re-assessment

Re-assessment is not available except when required by referral or deferral.

Notes: See Physics Assessment Conventions.

KIS Data Summary

Miscellaneous