MPhys Physics

The unit covers four main interlinked areas of mathematical physics: Sturm-Liouville Theory, Green’s Functions, Integral Equations and Calculus of Variations. All four sections have a dual focus: on the more formal properties of the equations, including their consequences for, for instance, the completeness of the eigenfunctions of Hermitian operators, but also on solving problems, including those with source terms, that occur in classical and quantum physics. Differential equations, which make the core of modern physics, will be discussed; their Green’s functions will be derived; the correspondence between differential and integral equations will be highlighted; Calculus of Variations will be elucidated.

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. These techniques include methods of solving ordinary differential equations, integral equations and problems of variational calculus.

On completion successful students will be able to:

1. Describe the basic properties of the eigenfunctions of Sturm-Liouville operators; derive the solutions in particular cases.
2. Recognize when a Green's function solution is appropriate and construct the Green's function for some well-known physical equations.
3. Recognize and solve particular cases of Fredholm and Volterra integral equations.
4. 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).

Two one hour, live in-person lectures per week where the core material with examples will be delivered. The recordings of these lectures will be on the podcast system. The lectures are accompanied by full notes and summaries online. This is augmented by weekly online short quiz questions with immediate solutions, and fortnightly sheets on in-depth problems, which are discussed in the examples classes. A Piazza discussion forum is also provided where students can ask questions with answers provided by other students and the unit lead. Formative feedback will be provided during example classes.

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

Reading list is given during the lectures.