Department of Physics, Unit Catalogue 2009/10 |
PH40073: Mathematical physics |
 Credits:
| 6 |
| Level:
| Masters |
| Period: | Semester 2 |
| Assessment:
| EX 100% |
| Supplementary Assessment: | Like-for-like reassessment (where allowed by programme regulations) |
| Requisites:
| Before taking this unit you must take PH10004 and take PH20029 and take PH30077 and take PH30025 |
| Description: | Aims: The aim of this unit is to develop students' understanding of some fundamental aspects of Physics, where a mathematical treatment is essential fully to appreciate the subject. For the section on phase transitions, the aim is for students to gain a quantitative understanding of the principles that govern first and second order phase transitions. For the section on relativity, the aim is for students to understand the use of four-vectors in relativistic electrodynamics, and to develop an understanding of curved space-time and the physical mathematics of general relativity. For the section on classical mechanics, the aim is for students to understand and apply the Lagrangian formulation of classical mechanics. Learning Outcomes: After taking the section on phase transitions the student should be able to: * perform mean field calculations of phase transitions; * define critical exponents and discuss scaling relations and universality classes; * describe the principles of real-space renormalisation; After taking the section on relativity the student should be able to: * solve problems in special relativity and relativistic electro-dynamics using four-vectors; * define, manipulate and interpret tensors which describe curved space-time; * derive and use the Schwarzchild metric to solve problems in general relativity; After taking the section on classical mechanics the student should be able to: * use the Lagrangian and Hamiltonian formulations to solve problems in classical mechanics; * use symmetries to derive conservation laws; * formulate and analyse equations of motion for systems of oscillators. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Of the three topics described below, it is anticipated that only two will be covered in any one year. Phase transitions: Phenomenology, classification of phase transitions. Mean field theories; Weiss theory, Landau theory, Van der Waals theory. Statistical mechanics of phase transitions; examples based on the Ising model. Introduction to scaling and the renormalisation group. Relativity: Special relativity. Lorentz transformation, four-vectors. Relativistic electrodynamics; electromagnetic field tensor, invariance of Maxwell's equations, Lorentz transformation of electric and magnetic fields. Introduction to general relativity; equivalence principle, curved spacetime and the mathematics of general relativity, Schwarzschild metric. Classical mechanics: Calculus of variations. Hamilton's principle, Lagrangian formulation of classical mechanics, examples. Symmetry and conservation laws. Linear and non-linear dynamics. Classical field theory. Non-linear wave equations. |