| Unit name | Advanced Quantum Theory |
|---|---|
| Unit code | MATHM0053 |
| Credit points | 20 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Dr. Muller |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) |
MATH35500 Quantum Mechanics (or alternatively the Quantum Mechanics included in compulsory units on programmes involving Physics) MATH21900 Mechanics 2 or MATH31910 Mechanics 23 (or alternatively 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 |
Why is this unit important?
This unit introduces a raft of methods that have become core to modern developments in Quantum Theory – path integrals, semiclassical approximations, perturbation theory via Feynman diagrams, and second quantisation. These ideas are essential for any graduate who intends to start research in Mathematical or Theoretical Physics as well as a range of other disciplines. A common theme behind some of the methods is to explore the connection between Quantum Mechanics and Classical Mechanics, which is both of fundamental importance and forms the basis of powerful approximation techniques. As an example for this connection, we will see that if the classical dynamics in a system is chaotic, this has surprising consequences for the quantum mechanical wavefunctions and the eigenvalues of the Hamiltonian. We expect that the unit will appeal to students with an interest in any of the areas where the methods are applied, including Quantum Field Theory where many of them originated historically.
How does this unit fit into your programme of study?
Advanced Quantum Theory is part of a range of 3rd and 4th year units that allow students to acquire a broad background and specialise in Quantum Theory (along with Quantum Mechanics, Quantum Information Theory, Quantum Computation, parts of Statistical Mechanics, and related units offered by the School of Physics). It also strengthens the link between this area and Mechanics 2/23. The elements dealing with Quantum Chaos have close connections to Random Matrix Theory (giving predictions for the behaviour of chaotic quantum systems) as well as Dynamical Systems and Ergodic Theory (introducing Classical Chaos from the point of view of Pure Mathematics).
An overview of content
Propagator and path integrals: We will introduce the propagator, a quantity describing time evolution in Quantum Mechanics, and show that it can be expressed as an integral with classical paths (including paths not satisfying the equations of motion) serving as integration variables. We will also discuss extensions of this idea.
Semiclassical approximations: In this section we will also see that the propagator can be approximated by a discrete sum over paths (now required to satisfy the equations of motion), and that many integrals relevant to Quantum Mechanics can be evaluated approximately using a stationary-phase approximation.
Quantum chaos: If the classical motion in a system is chaotic, this has important consequences for its quantum mechanical properties. Classical chaos means that, for example, classical trajectories with very similar initial positions and momenta quickly diverge from each other. This has surprising implications for the quantum mechanical wavefunctions as well as the energy levels. For example, the energy levels appear to ‘repel’ each other. We will use the links between classical and quantum mechanics established above to introduce elements of the theory of chaotic quantum systems.
Perturbation theory: Many systems have a Hamiltonian that can be written as an exactly solvable part plus a small perturbation. We will develop a technique that allows to approximately determine the propagator of such systems, with a visual representation using Feynman diagrams: This approach will be presented in the context of path integrals.
Many-particle systems and second quantisation: Finally, we will consider systems with several particles of the same type and discuss the form of wavefunctions for such systems. The Hamiltonian of many-particle systems can be expressed efficiently in terms of operators that create or annihilate particles in the system. Time permitting, we will discuss connections between this approach and path integrals.
How will students, personally, be different as a result of the unit
At the end of the unit the students will have improved their logical thinking, learnt to synthesise ideas from different disciplines, and have increased the range and scope of their problem-solving techniques. They will be able to apply a number of methods that are at the core of current research in Quantum Mechanics. Students considering research as a career option will have strengthened their preparation.
Learning Outcomes
Students will be able to
Teaching methods will include
Tasks which help you learn and prepare you for summative tasks (formative):
formative homework problems
Tasks which count towards your unit mark (summative):
90% timed examination, 10% assessed coursework
When assessment does not go to plan
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. If you have self-certificated your absence from an assessment (or, in certain scenarios, if you had failed the assessment), you will normally be required to complete it in the reassessment period. Please refer to your official results for the details of your reassessments. The Board of Examiners will take into account any exceptional circumstances and operates within the Regulations and Code of Practice for Taught Programmes.
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. MATHM0053).
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 assessment methods listed in this unit specification are designed to enable students to demonstrate the named learning outcomes (LOs). Where a disability prevents a student from undertaking a specific method of assessment, schools will make reasonable adjustments to support a student to demonstrate the LO by an alternative method or with additional resources.
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.
