Description

The mathematical techniques presented relate directly to the advanced modules at Stages 3 and 4 of Physics programmes, and also have wide applicability across the mathematical sciences. Practical skills are emphasised, rather than formal proofs.

Module Aims

This module aims to develop a deeper understanding of, and greater competence in using, some of the important mathematical methods and techniques of theoretical physics not covered in PHY2025.

Intended Learning Outcomes (ILOs)

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

• Module Specific Skills and Knowledge:
  1. find a Laurent expansion of a function of a complex variable; identify poles and branch points;
  2. evaluate a wide variety of definite integrals using the calculus of residues;
  3. use symmetry arguments when evaluating integrals;
  4. approximate the value of a definite integral using the method of steepest descent;
  5. deduce the first term in Stirling's expansion of the factorial function;
  6. approximate the value of the ground-state energy of a bound particle by using the variational method;
  7. find approximate solutions of the Shrödinger equation using the WKB method;
  8. identify the point group of a simple symmetric object;
  9. identify the 'bond', 'site' and continuum problems of percolation theory;
  10. solve problems using the theory of conformal mapping;
• Discipline Specific Skills and Knowledge:
  11. apply analytical and numerical skills in mathematics;
  12. formulate problems in a logical manner;
  13. adapt and apply the methods discussed in lectures to unseen problems;
• Personal and Key Transferable / Employment Skills and Knowledge:
  14. present and justify use of techniques and methods;
  15. work in groups - students are encouraged to work co-operatively together to solve assigned problems.

Syllabus Plan

1. Functions of a Complex Variable
   1. Revision of basic notations and properties of complex numbers
   2. Analytic functions
   3. Cauchy's theorem
   4. Laurent expansion
   5. Calculus of residues
2. Evaluation of Integrals
   1. Elementary methods
   2. Use of symmetry
   3. Contour integration
3. Conformal Mapping
   1. Theory
   2. Applications
4. Approximate Methods
   1. Method of steepest descent
   2. The WKB method
   3. Variational method in quantum mechanics
5. Special Chapters
   1. Elements of group theory
   2. Elements of percolation theory
   3. Non-linear Shrödinger equation (an example)

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:

• Mathews J. and Walker R.L. (1970), Mathematical Methods of Physics (2^nd edition), Benjamin, ISBN 0-805-37002-1
• Stauffer D. (1985), Introduction to Percolation Theory, Taylor and Francis, ISBN 0-850-663156

ELE:

• http://newton.ex.ac.uk/redirects/ELE/phy3062

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