|Delivery Type||Delivery length / details|
|Lecture||36 x 1-hour lectures|
|Seminars / Tutorials||8 x 1-hour tutorials|
|Workload Breakdown||(Every 10 credits carries a notional student workload of 100 hours.) Lectures and tutorials 44 hours Worksheets (6 x 5 hours) 30 hours Private study 123 hours Examination 3 hours|
|Assessment Type||Assessment length / details||Proportion|
|Semester Assessment||3 Hours Written Examination||100%|
|Supplementary Exam||3 Hours Written Examination||100%|
On completion of this module, students should be able to.
1. illustrate the basic theory and applications of operator theory;
2. define basic notions from operator theory and recognize their occurrence and relevance in applied problems;
3. solve specific problems from mathematical physics formulated in operator theoretic terms;
4. Perform algebraic and analytic computations based on operator techniques;
IMAPS wishes to introduce new level 3 modules reflecting research interests and expertise of new staff, thereby rectifying the previous problem of very limited range of modules for final year students. This module introduces the important field of operator theory. It is intended to offer this module in alternate years. Operator theory is only represented in the curriculum through spectral theory at present and we would like to introduce it as an option. It has critical importance to areas such as spectral analysis and quantum theory, both of which are strong research topics in the department. It introduces specific concepts central to open quantum systems, fundamental to understanding current research.
Modern quantum theory requires mathematical concepts and techniques going beyond traditional techniques encountered in standard textbooks. A proper understanding of these principals involves operator theoretic concepts, which will be presented in the module. The motivation is a description of open quantum systems.
Introduction to Hilbert spaces, definitions of bounded, unitary, projective and
self-adjoint operators. Applications of the spectral theorem and Stone's theorem.
Introduction of the notion of abstract operator algebras.
Applied Operator Theory
Motivating examples of operators from Mathematical Physics. Mathematical formulation of
Quantum Theory for closed dynamical systems.
Open Quantum Systems
Introduction to the quantum theory of measurement. Completely positive mappings.
Instruments and measurements. Lindblad's theory of dynamical semi-groups.
|Skills Type||Skills details|
|Application of Number||Throughout the module.|
|Communication||Students will be expected to submit written worksheet solutions|
|Improving own Learning and Performance||Feedback via tutorials|
|Information Technology||Extensive use of spreadsheets.|
|Personal Development and Career planning||Students will be exposed to an area of application that they have not previously encountered.|
|Problem solving||All situations considered are problem-based to a greater or lesser degree.|
|Research skills||Students will be encouraged to consult various books and journals for examples of application.|
|Subject Specific Skills||Using differential geometric techniques in modeling.|
This module is at CQFW Level 7