MPhys Physics with Theoretical Physics / Course details

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. 

On completion successful students will be able to:

Recognize when a Green's function solution is appropriate and construct the Green's function for some well-known physical equations.

Recognize and solve particular cases of Fredholm and Volterra integral equations.

Describe the basic properties of the eigenfunctions of Sturm-Liouville operators; derive the solutions in particular cases.

Solve a variational problem by constructing an appropriate functional, and solving the Euler-Lagrange equations.

Mathematical Methods for 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.

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.