Course Overview: This course builds on Part A Quantum Theory. The mathematical foundations of quantum theory are developed more deeply than in the Part A course, and general principles regarding the realisation of symmetries in quantum mechanical systems are developed. Systems of several particles are studied, including a consideration of identical particles and particle statistics. Along the way, a number of simple-but-relevant concepts from the theories of Lie groups, representation theory, and functional analysis are introduced in a hands-on fashion.
In semi-realistic systems, an exact solution to the quantum dynamics is rarely forthcoming. The second part of the course is largely devoted to developing several approximation techniques that can be applied in these more general settings. These are employed to study problems such as the determination of energy levels of the Helium atom.
The course concludes with a short introduction to scattering theory in one spatial dimension.
Learning Outcomes: Student will be able to state the postulates of nonrelativistic quantum theory and explain how they are realized in key examples. They will be able to analyse simple quantum systems that admit exact solutions by exploiting symmetries and algebraic techniques. They will be able to calculate approximations to energy levels, scattering states, and other properties of more complicated systems using perturbation theory, semi-classical techniques, and variational methods.
Course Synopsis: Abstract formulation of quantum mechanics in terms of linear operators on Hilbert spaces; Dirac notation; discrete and continuum states; time evolution and the propagator.
Systems of several particles and Hilbert space tensor products; distinguishable and indistinguishable particles; Fermi--Dirac and Bose--Einstein statistics; the Pauli exclusion principle; elementary aspects of quantum entanglement.
Symmetries in quantum mechanics as unitary and anti-unitary operators; rotations, angular momentum, and spin; spin-\(1/2\) and projective representations of \(\mathrm{SO}(3)\); addition of angular momentum; spin-statistics theorem; tensor operators and the Wigner--Eckart theorem.
Approximation methods: Rayleigh--Schrödinger perturbation theory; variational methods; WKB approximation and Bohr--Sommerfeld quantisation.
Elementary scattering theory in one dimension; relation between bound states and of poles/zeroes of the \(S\)-matrix.
