Application of Symmetry Groups in Physics

The course is given every other year, and will be given autumn 2027.

Transformations, generators, symmetries and conservation laws in quantum mechanics. Examples of groups. Conjugacy classes, subgroups, cosets, quotient groups, direct-product groups. Homomorphisms and isomorphisms. Representations and bases. Characters. Orthogonality relations and character tables for irreducible representations. Decomposition of reducible representations. Energy levels and degeneracy, perturbations and level splitting. Transfer and projection operators. Product representations. Continuous groups, Lie groups and Lie algebras. The rotation group and angular momentum. Irreducible tensor operators. Matrix elements and selection rules. Point groups. Space groups. Time reversal. Projective representations. Various applications, including crystal field splitting, optical spectra, and symmetry aspects of the band theory of crystals.

The course gives an introduction to group theory that will make a student able to analyze symmetries and their implications in a systematic and unified way, including solving or simplifying various problems in atomic, molecular and solid state physics for which symmetry plays a role. The most important specific learning outcomes are

(i) to know the connection between the energy levels of a Hamiltonian operator and the irreducible representations of its symmetry group, and be able to determine how the energy levels are split by perturbations, e.g. the splitting of atomic energy levels in crystal fields, and

(ii) to be able to determine conditions for when matrix elements of irreducible tensor operators (including the Hamiltonian) can be nonzero (so-called selection rules), with applications to e.g. optical absorption spectra.

A more general learning outcome is that the unified discussion of symmetries will give a deeper understanding of the structure of quantum mechanics.

Lectures and voluntary calculation exercises (in English and joint with FY3105).