Description

The emphasis in this module is on practical skills rather than formal proofs. Students will acquire skills in some key mathematical techniques that relate directly to the advanced modules they will meet in the later stages of their degree programme, but also have wide applicability across the mathematical sciences.

Module Aims

This module aims to enable the student to build on the knowledge and skills developed in PHY1026 in order to achieve a deeper understanding of and greater competence in some central mathematical ideas and techniques used throughout physics.

Intended Learning Outcomes (ILOs)

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

• Module Specific Skills and Knowledge:
  1. use probability theory to solve problems;
  2. calculate Fourier transforms and use them to solve problems
  3. solve partial differential equations by separation of variables;
  4. calculate eignvalues and eigenvectors and apply the the techniques to physical problems;
  5. use basis vectors to transform differential operator equations to matrix form and hence apply eigen equation techniques;
  6. obtain approximate solutions to differential equations through the use of perturbation theory;
  7. solve problems involving classical particles by applying the Lagrangian formulation classical mechanics;
  8. explain the calculus of variations and apply it to the solution of problems;
• Discipline Specific Skills and Knowledge:
  9. apply analytical and numerical skills in mathematics;
• Personal and Key Transferable / Employment Skills and Knowledge:
  10. formulate and tackle problems in a logical and systematic manner;
  11. present work clearly with justification of techniques and methods;
  12. work co-operatively with peers and with the demonstrators to solve guided problems.

Syllabus Plan

1. Probability theory
   1. Random variables
   2. Conditional probability
   3. Probability distributions
      1. Discrete
      2. Continuous
2. Lagrangian formulation of classical mechanics
   1. Calculus of variations
   2. Euler-Lagrange equations
3. Solution of linear partial differential equations
   1. Simple second order differential equations and common varieties: Harmonic oscillator, Schrödinger equation, Poisson's equation, wave equation and diffusion equation.
   2. Separation of variables: The Laplacian family of equations in physics, separation of variables, mechanics of the technique, form of solutions, general solutions in series form, relation to Fourier series, spatial boundary conditions, time dependence, initial conditions.
   3. Examples: rectangular drum, classical and quantum harmonic oscillator, waves at a boundary, temperature distributions, wavepacket/quantum particle in a box
   4. Role of symmetry: Cylindrical and spherical polar co-ordinates, appearance of special functions. Use of special functions by analogy to sin, cos, sinh, cosh etc.
   5. Examples: circular drum, hydrogen wave function
4. Linear Algebra
   1. Revision: Row and column vectors, matrices, matrix algebra, the solutions of systems of linear equations.
   2. Eigenvalue equations I: The matrix equation Ax=ax, solving the matrix equation, the secular determinant, eigenvalues and eigenvectors, canonical form, normal modes/harmonics, simple coupled oscillators.
   3. Eigenvalue equations II: Properties of eigenvectors: orthogonality, degeneracy, as basis vectors.
   4. Eigenvalue equations III: Differential equations as eigenvalue equations and the matrix representation Ax=ax; choosing the basis, solving the equation, the secular determinant, eigenvalues and eigenvectors.
   5. Examples: classical coupled modes, Schrödinger wave equation
   6. Approximate solutions to differential equations (perturbation theory): use of eigenvectors, first- and second-order through repeated substitution, problem of degeneracies.
   7. Examples: quantum particle in a well, a mass on drum, coupled particles

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:

• K.F. Riley, Hobson M.P. and Bence S.J. (2006), Mathematical Methods for Physics and Engineering: A Comprehensive Guide (3^rd edition), Cambridge University Press, ISBN 978-0521679718

Supplementary texts:

• Boas M.L. (2005), Mathematical Methods in the Physical Sciences (3^rd edition), John Wiley and Sons, ISBN 978-0-471-36580-8
• Gregory R.D. (2006), Classical Mechanics, Cambridge University Press, ISBN 0-521-534097
• James G. (1993), Advanced modern engineering mathematics, Addison-Wesley, ISBN 0-201-56519-6
• Kreyszig E. (2005), Advanced Engineering Mathematics (9^th edition), Wiley, ISBN 978-0-471-72897-9
• Spiegel M.R. (1971), Advanced Mathematics for Engineers and Scientists, Schaum Outline Series, McGraw-Hill, ISBN 0-070-60216-6
• Stroud K.A. and Booth D.J. (2013), Engineering Mathematics (7^th edition), Paulgrave MacMillan, ISBN 978-1-137-03120-4

ELE:

• http://newton.ex.ac.uk/redirects/ELE/phy2025

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