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Advanced Quantum Mechanics - PHY00048M

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  • Department: Physics
  • Module co-ordinator: Prof. Aires Ferreira
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

This course introduces key advanced topics in quantum mechanics that bridge the gap between earlier courses and physics research.

Related modules

 Pre-requisites: Stage 2 Quantum II AND Stage 2 Mathematics II AND Stage 3 Quantum III

Module will run

Occurrence Teaching period
A Semester 2 2024-25

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.

  • To study the quantum theories of angular momentum and scattering, and the role of symmetries and the algebraic approach in quantum mechanics.

Module learning outcomes

  • Illustrate the relation between symmetries and conservation laws.

  • Calculate the time-dependence of a wavefunction, and its consequences for physical observables.

  • Derive and apply the results of time-dependent perturbation theory up to first order.

  • Calculate geometric phases for adiabatically driven quantum systems.

  • Explain and apply the laws of quantum mechanics for many-particle systems.

  • Deduce and apply the general theory of angular momentum.

Module content

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

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 semimetals 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.

Angular momentum theory: Symmetries and rotations; angular momentum multiplets (Ladder operators); addition of angular momenta and selection rules including Parity (Clebsch-Gordan coefficients and the Wigner-Eckart theorem).

Quantum states of the harmonic oscillator: Creation and annihilation operators; coherent states and squeezed states.

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Advanced Quantum Mechanics		 3 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Advanced Quantum Mechanics		 3 hours 100

Module feedback

'Feedback’ at a university level can be understood as any part of the learning process which is designed to guide your progress through your degree programme. We aim to help you reflect on your own learning and help you feel more clear about your progress through clarifying what is expected of you in both formative and summative assessments.

A comprehensive guide to feedback and to forms of feedback is available in the Guide to Assessment Standards, Marking and Feedback. This can be found at:

https://www.york.ac.uk/students/studying/assessment-and-examination/guide-to-assessment/

The School of Physics, Engineering & Technology aims to provide some form of feedback on all formative and summative assessments that are carried out during the degree programme. In general, feedback on any written work/assignments undertaken will be sufficient so as to indicate the nature of the changes needed in order to improve the work. Students are provided with their examination results within 25 working days of the end of any given examination period. The School will also endeavour to return all coursework feedback within 25 working days of the submission deadline. The School would normally expect to adhere to the times given, however, it is possible that exceptional circumstances may delay feedback. The School will endeavour to keep such delays to a minimum. Please note that any marks released are subject to ratification by the Board of Examiners and Senate. Meetings at the start/end of each semester provide you with an opportunity to discuss and reflect with your supervisor on your overall performance to date.

Our policy on how you receive feedback for formative and summative purposes is contained in our Physics at York Taught Student Handbook

Indicative reading

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

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

LD Landau and EM Lifshitz: Quantum Mechanics - Non-relativistic Theory (Butterworth Heinemann)

A Messiah: Quantum Mechanics (Dover)

AIM Rae: Quantum mechanics (Taylor and Francis)

JJ Sakurai and JJ Napolitano: Modern Quantum Mechanics (Pearson)

S Weinberg: Lectures on quantum mechanics (Cambridge)


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.