Occurrence | Teaching cycle |
---|---|

A | Autumn Term 2019-20 to Spring Term 2019-20 |

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:

**1. Advanced Quantum Mechanics:** To study the consequences of the time-dependence of the wavefunction in quantum mechanics, the emergence of the basic laws of classical mechanics from quantum mechanics, the quantum mechanics of many-particle systems, and second quantisation.

**2. Further Quantum Mechanics: **To study the quantum theories of angular momentum and scattering, and the role of symmetries and the algebraic approach in quantum mechanics.

**Advanced Quantum Mechanics**

- 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 to selection rules for atomic transitions.
- Explain the origin of the laws of classical mechanics 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 blackboard during lectures, and will be supplemented by a one-page hand-out distributed on paper. This hand-out, together with audio recordings of lectures and a record of problems set, lecture rescheduling and similar information, will be made available through the VLE.

**Further Quantum Mechanics**

- Illustrate the relation between symmetries and conservation laws.
- Deduce and apply the general theory of angular momentum.
- Deduce and apply the Born approximation and the method of partial waves in potential scattering theory.
- Apply creation and annihilation operators of the harmonic oscillator.
- Construct solutions to complex unseen problems in all of the aforementioned topics.

Comprehensive lecture notes should be taken down from the blackboard during lectures. Supplementary notes will be provided and made available through the VLE.

**Syllabus**

__Advanced Quantum Mechanics__

**T ime-dependence:** Brief review of time-dependent Schrödinger equation; stationary states; time-evolution of general wavefunctions; time evolution operator; time-energy uncertainty relation. Time-dependent perturbation theory. Fermi's golden rule; selection rules for atomic transitions re-examined. Ehrenfest’s theorem. [5 lectures]

** The classical limit:** Classical mechanics of particles as a limit of quantum mechanics, mostly studied through wavepacket motion. [2]

** Many-particle systems:** 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. Density-functional theory and the local-density approximation. [5]

* Second quantisation:* Creation, annihilation and number operators; their use for many-particle systems; anti- commutation relations; field operators; Heisenberg picture. Introduction to many-body perturbation theory. Introduction to quantisation of the electromagnetic field. [6]

__Further Quantum Mechanics__

** Symmetries and angular momentum:** 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).

** Potential scattering:** Lippmann-Schwinger equation, scattering amplitudes and the Born approximation • Partial waves, phase shifts and resonances.

** Quantum states of the harmonic oscillator:** Creation and annihilation operators • Coherent states and squeezed states.

Task | Length | % of module mark |
---|---|---|

University - closed examinationAdvanced Quantum Mechanics |
1.5 hours | 50 |

University - closed examinationFurther Quantum Mechanics |
1.5 hours | 50 |

None

Task | Length | % of module mark |
---|---|---|

University - closed examinationAdvanced & Further Quantum Mechanics |
3 hours | 100 |

Marks for the individual exams received from supervisor. Detailed model answers provided on the intranet.

**Advanced Quantum Mechanics**

Rae A I M: *Quantum mechanics* (Taylor & Francis)***

Merzbacher E: *Quantum mechanics* (Wiley, 1998) **

Schiff L I: *Quantum mechanic*s (McGraw-Hill) **

Ziman J M:* Elements of advanced quantum theory* (CUP)*

**Further Quantum Mechanics**

Weinberg S:* Lectures on quantum mechanics *(Cambridge, 2013)

Sakurai J J:* Modern quantum mechanics* (Addison Wesley, 1994)

Messiah A: *Quantum Mechanics Volume II *(Dover, 1999)

Landau L D and Lifshitz E M:* Quantum Mechanics *(Non-relativistic Theory) (Butterworth-Heinemann, 1977)