MPhys Physics

Mathematical Methods for Physics

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.

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.

2. 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).

3. 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).

4. 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).

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

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)