BSc Physics with Theoretical Physics

Year of entry: 2024

Course unit details:
Mathematical Methods for Physics

Course unit fact file
Unit code PHYS40672
Credit rating 10
Unit level Level 4
Teaching period(s) Semester 2
Available as a free choice unit? No


Mathematical Methods for Physics


Unit title Unit code Requirement type Description
Mathematics of Waves and Fields PHYS20171 Pre-Requisite Compulsory
Lagrangian Dynamics PHYS20401 Pre-Requisite Compulsory
Mathematical Fundamentals of Quantum Mechanics PHYS30201 Pre-Requisite Compulsory
Complex Variables and Vector Spaces PHYS20672 Pre-Requisite Compulsory


The aim of this course is to achieve an understanding and appreciation, in as integrated a form as possible, of some mathematical techniques which are widely used in theoretical physics.

Learning outcomes

On completion successful students will be able to:

  1. Describe the basic properties of the eigenfunctions of Sturm-Liouville operators.
  2. Derive the eigenfunctions and eigenvalues of S-L operators in particular cases.
  3. Recognize when a Green's function solution is appropriate and construct the Green's function for some well-known physical equations.
  4. Recognize and solve particular cases of Fredholm and Volterra integral equations.
  5. Solve a variational problem by constructing an appropriate functional, and solving the Euler-Lagrange equations.


  1. Ordinary differential equations and Sturm-Liouville theory (8 lectures)

Linear second-order ODEs: singular points, boundary conditions. Hermitian Sturm-Liouville operators: properties of eigenvalues and eigenfunctions. Orthogonal and generalised-orthogonal polynomials. Generating functions, recurrence relations, series solutions. Fourier and Laplace transform methods.   Recap of special functions.

  1. Green's functions (6 lectures)

Definition. Example: electrostatics.  Construction of Green's functions: the eigenstate method; the continuity method.  Initial-value problems and causality. Partial differential equations: The Fourier transform method; retarded Green’s functions. Quantum scattering in the time-independent approach and Born approximation (perturbation theory).

  1. Integral equations (5 lectures)

Classification: integral equations of the first and second kinds; Fredholm and Volterra equations. Simple cases: separable kernels; equations soluble by Fourier transform; problems reducible to a differential equation.  Eigenvalue problems: Hilbert-Schmidt theory, resolvant kernel. Neumann series solution (perturbation theory).

  1. Calculus of variations (5 lectures)

Recap of Functionals: stationary points and the Euler-Lagrange equation; the functional derivative. Constrained variational problems; Lagrange's undetermined multipliers. The isoperimetric problems. The catenary. Variable end-points. The Rayleigh-Ritz method. The completeness theorem for Hermitian Sturm-Liouville operators (if time).

Assessment methods

Method Weight
Written exam 100%

Feedback methods

Feedback will be available on students’ individual written solutions to examples sheets, which will be marked, and model answers will be issued.

Recommended reading

Arfken, G.B. Weber, H.J. Mathematical Methods for Physicists (Academic Press)
Riley, K.F. Hobson, M. P. & Bence, S. J. Mathematical Methods for Physics and Engineering (CUP)

Study hours

Scheduled activity hours
Assessment written exam 1.5
Lectures 23
Independent study hours
Independent study 75.5

Teaching staff

Staff member Role
Judith McGovern Unit coordinator

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