PHY3140 Methods of Theoretical Physics

2007-2008

Code: PHY3140
Title: Methods of Theoretical Physics
Instructors: Dr M.E. Portnoi
CATS credits: 10
ECTS credits: 5
Availability: unrestricted
Level: 3
Pre-requisites: N/A
Co-requisites: N/A
Background Assumed: Mathematics for Physicists (PHY1116)
Duration: Semester II
Directed Study Time: 22 lectures
Private Study Time: 78 hours
Assessment Tasks Time: -
Observation report: 2004/05 TJH

Aims

This module will help students develop a deeper understanding of, and greater competence in using, some of the important mathematical methods and techniques of theoretical physics not covered in PHY2018. Practical skills, rather than formal proofs, will be emphasised. Students will acquire skills in techniques that relate directly to the advanced modules at levels 3 and 4, and which also have wide applicability across the mathematical sciences.

Intended Learning Outcomes

After completing this module, the student should be able to:

Module Specific Skills

• find a Laurent expansion of a function of a complex variable; identify poles and branch points;
• evaluate a wide variety of definite integrals using the calculus of residues;
• use symmetry arguments when evaluating integrals;
• approximate the value of a definite integral using the method of steepest descent;
• deduce the first term in Stirling's expansion of the factorial function;
• approximate the value of the ground-state energy of a bound particle by using the variational method;
• find approximate solutions of the Shrödinger equation using the WKB method;
• identify the point group of a simple symmetric object;
• identify the 'bond', 'site' and continuum problems of percolation theory;

Discipline Specific Skills

• apply analytical and numerical skills in mathematics;
• formulate problems in a logical manner;

Personal and Key Skills

• present and justify use of techniques and methods;
• work in groups - students are encouraged to work co-operatively together to solve assigned problems.

Learning and Teaching Methods

Lectures, handouts, problem classes, solutions to problem sheets, e-learning resources.

Assignments

Problem sheets (distributed weekly)

Assessment

One 90-minute examination (100%).

Syllabus Plan and Content

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. Approximate Methods
   1. Method of steepest descent
   2. The WKB method
   3. Variational method in quantum mechanics
4. Special Chapters
   1. Elements of group theory
   2. Elements of percolation theory

Core Text

Arfken G.B. and Weber H.J. (2001), Mathematical methods for physicists (5^th edition), Academic Press, ISBN 0-120-59826-4 (UL: 510 ARF)

Supplementary Text(s)

Mathews J. and Walker R.L. (1970), Mathematical Methods of Physics (2^nd edition), Benjamin, ISBN 0-805-37002-1 (UL: 530.15 MAT)
Stauffer D. (1985), Introduction to Percolation Theory, Taylor and Francis, ISBN 0-850-663156 (UL: 530.13 STA)

Formative Mechanisms

Progress in the set problems will be reviewed weekly.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.