Advanced Quantum Mechanics - PHY00019M

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Module summary

Key advanced topics in quantum mechanics that bridge the gap between earlier courses and physics research

Related modules

Additional information

To enrol on this module, students will need to have taken Quantum Mechanics courses in Stages 1-3 Physics, or the equivalent

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

The overall aim of the module is to develop in students a knowledge of key advanced topics in quantum mechanics that bridge the gap between earlier courses and physics research. Specifically:

To study the consequences of the time-dependence of the wavefunction in quantum mechanics, the adiabatic evolution of quantum states and the emergence of the Berry phase, the quantum mechanics of many-particle systems, and second quantisation.

 

Module learning outcomes

• ·         Calculate the time-dependence of a wavefunction, and its consequences for observables.
• ·         Derive and apply the results of time-dependent perturbation theory up to first order.
• ·         Derive and apply Fermi's golden rule, and explain the relevance of selection rules for atomic transitions and opto-electronic phenomena in solids.
• ·         Explain the origin of the Berry phase using simple calculations of the types given in lectures.
• ·         Explain and apply the laws of quantum mechanics for many-particle systems and the main techniques used to study their implications.
• ·         Derive the main results of second quantisation.
• ·         Describe, and apply to unseen problems, all the topics in the syllabus.

Comprehensive lecture notes should be taken down from the slides during the video lectures, and will be supplemented by summary notes and handouts distributed via PDF format. These documents, together with interactive applets and the record of problems set, lecture rescheduling and similar information, will be made available through the VLE.

Module content

Syllabus

Operator methods, the classical limit, and symmetries: Brief review of Dirac notation; state vector; observables; Ehrenfest theorem; the classical limit. Introduction to symmetries.

Time-dependence: Brief review of Schrödinger equation; stationary states; time-evolution of general wavefunctions. Time-dependent perturbation theory. Fermi's golden rule. Dyson series. Introduction to Feynman diagrams.

Geometrical phase and topology: Aharonov–Bohm effect of charge particles; adiabatic cyclic evolution and Berry phase of quantum systems; monopoles of Berry curvature. Introduction to topological insulators and Weyl-Dirac semi-metals in 2 and 3 dimensions.

Many-particle systems and second quantisation: Identical particles and exchange symmetry, fermions and bosons, the Pauli Principle; use of Slater determinants. Variational principle for many-electron systems; the Hartree and Hartree-Fock approximations. Creation, annihilation and number operators; their use for many- particle systems; anti-commutation relations; field operators; Heisenberg picture. Introduction to many-body perturbation theory.

Indicative assessment

Task	% of module mark
Online Exam - 24 hrs (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Online Exam - 24 hrs (Centrally scheduled)	100

Module feedback

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Indicative reading

Griffiths, DJ, Introduction to Quantum Mechanics (CUP)

Rae, AIM, Quantum Mechanics (Taylor and Francis)

Sakurai, JJ.: Advanced Quantum Mechanics (Addison-Wesley)

Ballentine, LE: Quantum Mechanics: A Modern Development (World Scientific)

Weinberg S: Lectures on Quantum Mechanics (CUP)

Bernevig, BA & Hughes, TL: Topological Insulators and Topological Superconductors (PUP)