Classical and Quantum Dynamics
| Code | School | Level | Credits | Semesters |
| MATH3035 | Mathematical Sciences | 3 | 20 | Spring UK |
- Code
- MATH3035
- School
- Mathematical Sciences
- Level
- 3
- Credits
- 20
- Semesters
- Spring UK
Summary
The course introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics. The course explores how classical concepts enter quantum mechanics, and how they can be used to find approximate semiclassical solutions.
In classical dynamics we discuss full integrability and basic notions of chaos in the framework of Hamiltonian systems, together with advanced methods like canonical transformations, generating functions and Hamiltonian-Jacobi theory.
In quantum mechanics we recall Schrödinger’s equation and introduce the semiclassical approximation. We derive the Bohr-Sommerfeld quantization conditions based on a WKB-approach to the eigenstates. We will discuss some quantum signatures of classical chaos and relate them to predictions of random-matrix theory.
We will also introduce Gaussian states and coherent states and discuss their semiclassical dynamics and how it is related to the corresponding classical dynamics. An elementary introduction to complete descriptions of quantum mechanics in terms of functions on the classical phase space will be given.
Target Students
Single Honours and Joint Honours students from the School of Mathematical Sciences including Mathematical Physics students.
Classes
- Two 2-hour lectures each week for 12 weeks
Assessment
- 100% Exam 1 (3-hour): Written examination
Assessed by end of spring semester
Educational Aims
Thiscourse forms part of the Quantum Theory pathway within Mathematical Physics. It introduces advanced concepts and methods used in analysis of models employed for understanding of behaviour of classical and quantum mechanical systems and how the two are related. The methods introduced in thiscourse have applications in a wide range of topics in applied mathematics and mathematical physics.Learning Outcomes
A student who completes this module successfully will be able to:
- L1 - apply some advanced methods of Mathematical Physics such as Gaussian integration, Poisson resummation, the saddle-point approximation, and asymptotic analysis;
- L2 - apply advanced methods of Hamiltonian Mechanics (canonical transformations, generating functions, action-angle variables, Hamilton-Jacobi theory) in the context of integrable and chaotic classical dynamics;
- L3 - apply semiclassical approximations for the eigenfunctions and eigenvalues of a quantum Hamiltonian based on the WKB method (Bohr-Sommerfeld quantization) and justify the validity of the approximations;
- L4 - apply semiclassical approximations to the dynamics of quantum states;
- L5 - state and apply elementary facts about Gaussian and coherent states, and about phase space formulations of quantum mechanics.