MMath&Phys Mathematics and Physics

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This unit introduces students to some of the more advanced concepts and techniques of modern quantum mechanics, and thus acts as a bridge into diverse research fields such as quantum information and computation, quantum optics, condensed matter theory, and nuclear and particle theory.  

The unit will first recap and extend students’ knowledge of the mathematical structures of quantum mechanics, introducing symmetries, unitary operators, and conservation laws.  

The coupling of charged quantum mechanical particles to electromagnetic fields will then be developed, including a discussion of the gauge principle in quantum mechanics and coupling to magnetic fields. The basic principles of non-relativistic quantisation of the electromagnetic field will be introduced, highlighting the main differences to coupling to classical electromagnetic fields.

A more formal treatment of angular momentum in quantum mechanics will be covered, including Clebsch-Gordan coefficients and vector operators. Non-degenerate and degenerate perturbation theory will be developed and applied, for example to the fine structure of hydrogen. Further approximation approaches will be formulated and applied to a wider range of both time-independent and time-dependent problems. 

Follow - Up Units

PHYS40481 - Quantum Field Theory

PHYS40682 - Gauge Theories

To enhance knowledge and understanding of quantum mechanics, in particular its underpinning mathematical structures, and to prepare students for applications encountered in Quantum Field Theory, Gauge Theories, Quantum Optics, and Quantum Matter. 

On the successful completion of the course, students will be able to:  

ILO 1

Define and apply the mathematical underpinnings and symmetry operations of quantum mechanics.

ILO 2

Work with the algebra of angular momentum operators and their eigenvalues to solve problems in quantum mechanics, including the addition of angular momenta.  

ILO 3

Derive a mathematical description of quantum motion in electromagnetic fields.

ILO 4

Use both time-independent and time-dependent perturbation theory to find approximate solutions to problems in quantum mechanics.

1.    Symmetries in quantum mechanics 

Rotations, space-time reflections and parity

Unitary operators for space and time translations

Conversation laws

Schrödinger vs Heisenberg picture

2.    Time-dependent perturbation theory 

Fermi's Golden Rule

Selection rules for atomic transitions

Emission and absorption of radiation

Finite width of excited state

Selection rules for hydrogen

3.    Coupling to E&M fields 

Minimal coupling

Landau levels

The Gauge Principle in Quantum Mechanics

The Pauli-Schrödinger equation

4.     Relativistic wave equations 

The Klein-Gordon equation

The Dirac equations

Chirality and helicity

Lorentz invariance and the non-relativistic limit

The hydrogen atom and fine structure

Graphene

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 made available via the course online page. The lectures will be accompanied by online lecture notes and fortnightly exercise sheets. A Piazza discussion forum will also be provided where students can ask questions with answers provided by other students and the unit lead. 

Feedback will be provided via solutions to the problem sheets, which will be made available electronically on Blackboard. More detailed feedback will be provided through examples classes which are integrated within the 24 lectures.

R. Shankar, Principles of Quantum Mechanics, 2nd edition (Springer, 1994).

J. Binney and D. Skinner, The Physics of Quantum Mechanics (OUP, 2014).

J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd edition (CUP, 2020).  

S. Gasiorowicz, Quantum Physics, 3rd edition (Wiley, 2003).