BSc Physics with Theoretical 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. 

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. Mathematical structure of quantum mechanics (linear algebra recap). (2 lectures)
2. Symmetries in quantum mechanics: Rotations, space-time reflections and parity, Unitary operators for space and time translations, Conservation laws, Schrödinger vs Heisenberg picture, Ehrenfest theorem. (4 lectures)
3. Coupling to E&M fields: Minimal coupling, Landau levels, The Gauge Principle in Quantum Mechanics, The Pauli-Schrödinger equation. Quantization of the EM field. (6 lectures)
4. Angular Momentum: recap of general properties of angular momentum, Addition of angular momentum, Clebsch-Gordan coefficients, vector operators. (3 lectures)
5. Time-independent non-degenerate and degenerate perturbation theory, The fine structure of hydrogen: External fields: Zeeman and Stark effect in hydrogen. (4 lectures)
6. Time-dependent perturbation theory: Interaction picture, Fermi's Golden Rule, Emission and absorption of radiation, Selection rule proofs for hydrogen, Spontaneous emission, Finite width of excited state. (3 lectures)

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).