| Unit name | Advanced Quantum Theory |
|---|---|
| Unit code | MATHM0013 |
| Credit points | 10 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1B (weeks 7 - 12) |
| Unit director | Dr. Muller |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) |
MATH10015 Linear Algebra, MATH10012 ODEs, Curves and Dynamics, MATH35500 Quantum Mechanics Plus either MATH21900 Mechanics 2 or MATH31910 Mechanics 23 or PHYS30008 Analytical Mechanics. |
| Units you must take alongside this one (co-requisite units) |
None |
| Units you may not take alongside this one |
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 concepts leading up to 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 mathematical or theoretical physics as well as range of other disciplines.
Unit Description
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. 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.
The course starts by introducing path integrals. These provide a way to describe quantum mechanical time evolution in terms of classical trajectories. Crucially, the integration runs over all trajectories with a given initial and final point including those that do not satisfy the classical laws of motion. Path integrals for simple systems such as the harmonic oscillator will be computed exactly. We will then introduce perturbation theory and Feynman diagrams, which provide a powerful method to approximately evaluate path integrals of more complicated systems. We will also generalise the path integral formalism to many-particle systems. To do this we will first introduce second quantisation, a formalism to study many-particle systems that is technically similar to the treatment of the harmonic oscillator in terms of raising and lowering operators. Then this approach will be connected to the path integral formalism. Here the treatment of fermionic many-particle systems requires particular attention as the corresponding path integral has to be formulated in terms of anticommuting (Grassmannian) variables. In this context we will also discuss the important concept of supersymmetry.
This unit is also part of the Oxford-led Taught Course Centre (TCC), and can be taken by PhD students in Bristol and (via videolink) in our TCC partner departments. For MSci and MSc students the unit will have the usual level for a fourth year course.
Relation to Other Units
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.
A student successfully completing this unit will be able to
Transferable Skills
The schedule is slightly different from other courses. There will be a weekly two-hour slot.
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. With 15 lectures the course will thus run over eight weeks.
In addition there will be one-hour problem classes, not transmitted over the internet, in about four of these weeks. This will be complemented by lecture notes, problem sheets, and a revision class.
90% Timed, open-book examination
10% Coursework
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.
If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.
If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM0013).
How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours
of study to complete. Your total learning time is made up of contact time, directed learning tasks,
independent learning and assessment activity.
See the University Workload statement relating to this unit for more information.
Assessment
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit.
The Board considers each student's outcomes across all the units which contribute to each year's programme of study. For appropriate assessments, if you have self-certificated your absence, you will normally be required to complete it the next time it runs (for assessments at the end of TB1 and TB2 this is usually in the next re-assessment period).
The Board of Examiners will take into account any exceptional circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.
