Outline:

This module builds on the foundations laid in PHAS0022 Quantum Mechanics. It aims to extend the students' knowledge base and to give a deeper understanding of the subject. The course material is essential for many courses offered in the MSci year. The course gives a formal exposition of quantum mechanics in the form of matrix mechanics using Dirac notation. Topics include: Dirac Notation; Linear harmomic oscillator by operator techniques; Theory of orbital, spin and generalised angular momentum, with an introduction to coupling of two angular momenta, full treatment; time-independent perturbation theory to second-order including the degenerate perturbation theory, the variational principle, systems of identical particles; Pauli principle, bosons and fermions.

Aims:

This course aims to:

Introduce the basic concepts of Heisenberg’s matrix mechanics. The second year course (PHAS0022) dealt primarily with the Schrödinger’s matter wave dynamics. In PHAS0042, matrix mechanics is introduced as an alternative approach to quantum dynamics. It is also shown to provide a complementary approach, enabling the treatment of systems (such as spin systems) where solutions of the non-relativistic Schrödinger wave equation in position coordinates is not possible
	Apply matrix mechanics and operator algebra to the Quantum Harmonic Oscillator and its relation to the 2nd year wave dynamics solutions using Hermite polynomials
	Apply matrix mechanics and operator algebra to quantum angular momentum
	Demonstrate that matrix mechanics predicts and permits solution of spin-1/2 systems using Pauli matrices
	Develop understanding of fundamental concepts using these new methods. The Heisenberg uncertainty principle is shown to be just one among a family of generalized uncertainty relations, which arise from the basic mathematical structure of quantum theory, complementing arguments (eg Heisenberg microscope) introduced in the second year
	Explore some concepts of two-particle systems. The addition of two-spins is analysed including fundamental implications, exemplified by the Einstein-Podolsky-Rosen paradox and Bell inequalities
	Introduce approximate methods (time-independent perturbation theory, variational principle) to extend the PHAS0022 analytical solution of the hydrogen atom to encompass atoms in weak external electric and magnetic fields and two-electron systems like helium atoms
	Introduce symmetry requirements and the Pauli Principle
Teaching and Learning Methodology:

This course is delivered via weekly lectures supplemented by a series of workshops and additional discussion.

In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.