Unit name | Advanced Quantum Theory |
---|---|

Unit code | MATHM0013 |

Credit points | 10 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 2C (weeks 13 - 18) |

Unit director | Dr. Muller |

Open unit status | Not open |

Pre-requisites |
MATH11005 (Linear Algebra and Geometry), MATH11006 (Analysis 1), MATH 11007 (Calculus 1), either MATH 21900 (Mechanics 2) or MATH 31910 (Mechanics 23), MATH 35500 (Quantum Mechanics), or comparable units. Students will be expected to have attained a degree of mathematical maturity and facility at least at the level of a beginning Level M/7 student. |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Unit aims

The aims of this unit are to introduce and develop some key ideas and techniques of modern quantum theory. These ideas – functional integration, perturbation theory via Feynman diagrams and supersymmetry – are central concepts with extremely wide applicability within modern physics. The aim is to introduce the ideas and also to enable the student to be able to do example calculations with these sophisticated tools. This unit provides essential techniques for any graduate who intends to start research in theoretical or mathematical physics as well as range of other disciplines.

General Description of the Unit

Quantum theory is the fundamental framework within which a vast section of modern physics is cast: this includes atomic, molecular and particle physics as well as condensed matter and statistical physics, and modern quantum chemistry. In recent years it has also had unexpected and deep impact on pure mathematics. Fundamental to applying quantum theory in these areas are the more sophisticated techniques and ideas introduced in this course, namely path integrals, perturbation theory via Feynman diagrams and supersymmetry. These ideas not only allow quantum theory to be applied to these areas but also introduce a raft of concepts which have become a standard language for these fields.

NOTE: This unit is also part of the Oxford-led Taught Course Centre (TCC), and is taken by first- and second-year PhD students in Bristol and its TCC partner departments. The unit has been designed primarily with a postgraduate audience in mind. Undergraduate students should not normally take more than one TCC unit per semester.

Relation to Other Units

Quantum Mechanics and Mechanics 2/23 or equivalent units are prerequisites.

The methods introduced in this course are used in current research in several areas of mathematical and theoretical physics. Units giving an introduction into some of these areas are Statistical Mechanics, Quantum Information, Quantum Chaos, and Random Matrix Theory in Mathematics, and Relativistic Field Theory as well as several courses dealing with Condensed Matter in Physics. The Physics unit Advanced Quantum Physics includes complementary material about the Feynman path integral outside a field theoretical context.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Learning Objectives

A student successfully completing this unit will be able to:

- construct and use of the classical action for particles and fields;
- write down the quantum mechanical path integral for particles in 1d and be able to compute it for simple cases;
- construct and use of the functional integral for quantum field theory and statistical mechanics;
- compute functional integrals for free theories using the key tool: Wicks theorem;
- write the perturbation theory for interacting theories using the Feynman diagram expansion;
- define Grassmann variables and describe their properties;
- perform integration over Grassmann variables;
- describe the concept of supersymmetry quantum mechanics and explain how it plays out in simple examples;
- use ideas from supersymmetry to compute functional integrals;
- appreciate how the subject relates to some other areas of mathematics and physics, including, for example, statistical physics, particle physics, solid state physics and quantum information theory; be able to apply results from the course to problems in these areas.

Transferable Skills

- Clear, logical thinking.
- Problem solving techniques.
- Assimilation and use of complex and novel ideas.
- Appreciating connections between and unifying principles behind different areas of research.

The unit will be delivered through lectures. The lectures will be transmitted over the internet as part of the Taught Course Centre (TCC). The TCC is a consortium of five mathematics departments, including Bath, Bristol, Imperial College, Oxford and Warwick.

The lectures will comprise 15 hrs in total, at not more than 2 hrs per week.

In addition there will be problem sheets and about 3 problem and revision classes.

100% Examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

- Condensed matter field theory, A Altland and B Simons. 2nd ed (Cambridge University Press, 2010)
- Quantum Field Theory in a Nutshell, A Zee (Princeton University Press, 2003)
- Quantum Mechanics and Path Integrals: Emended Edition, RP Feynman, AR Hibbs and DF Styer (Dover, 2010)
- Quantum signatures of chaos, F Haake. 2nd rev (Springer, 2001)
- Path integral methods in quantum field theory, R Rivers (Cambridge University Press, 1987)