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# Foundations of Mathematical Physics

7CCMMS30

7

Autumn

15

## Module description

Aims

This course reviews the main concepts at the foundation of mathematical physics. It is a course that will provide a strong, basic mathematical knowledge to you upon which all other modules of the MSc programme can rely. It covers aspects of group theory, special relativity, classical mechanics, quantum mechanics, special functions and complex analysis.

Syllabus

Group theory: Fundamental concepts in group theory and representation theory  will be covered starting from the axioms defining a group and focussing on groups of interest to theoretical physics such as the Lorentz group, SO(3) and SU(2).

Special Relativity: The Lorentz transformations will be derived andLorentz tensors introduced. The generators of the Lorentz group will be found and four vectors and Lorentz invariants will be covered in detail. The Lie algebra isomorphism Lie(SO(1,3))= Lie(SU(2))+Lie(SU(2)) will be shown and the Poincare group discussed.

Mechanics: Lagrangian mechanics; configuration space; Maxwell's electromagnetism from a Lagrangian; Noether's theorem and examples; the action principle; Hamiltonian mechanics; Poisson brackets and phase space. The example of the simple harmonic oscillator will be developed.

Quantum Mechanics: Quantisation; Hilbert space; observables and self-adjoint operators; eigenvalues and eigenvectors; discrete and continuous bases for the Hilbert space; Bra-ket notation; commuting operators; the probabilistic interpretation; the Dirac delta "distribution"; transformation between different bases of the Hilbert space; the Schrödinger equation; the Heisenberg and Schrödinger pictures; the Probability current; quantisation of the harmonic oscillator.

Special Functions: Green's function; Legendre polynomials; Hermite polynomials and the harmonic oscillator.

Complex Functions: Riemann surfaces and mutli-valued functions; complex differentiation; harmonic functions in the plane; residues; complex integration; and poles on the contour.

Teaching arrangements

Eight hours a week for the first two weeks and four hours a week for the third and fourth weeks of the autumn term

Prerequisites

None

Formative assessment

Homework will be given out and the solutions will be provided. Difficulties with the problems will be explained during the module.

## Assessment details

Written examination

## Educational aims & objectives

This course reviews the main concepts at the foundation of mathematical physics. It is a course that will provide a strong, basic mathematical knowledge to you upon which all other modules of the MSc programme can rely. It covers aspects of group theory, special relativity, classical mechanics, quantum mechanics, special functions and complex analysis.

## Teaching pattern

Eight hours a week for the first two weeks and four hours a week for the third and fourth weeks of the autumn term