This module constructs a bridge between macroscopic thermodynamics and microscopic Statistical Mechanics. Using mathematical methods and fundamental laws of physics for individual particles, students acquire a strong understanding of Classical Statistical Mechanics at an advanced level, and also introductory knowledge of quantum Statistical Physics.
The aim of the course is to give students an introduction to the fundamental aspects of Solid State Physics. It is a prerequisite for all the courses to be followed in the Condensed Matter Physics second year Masters program. It is an elective course for all other Physics Masters. Topics covered include chemical bonds, crystals, phonons, electronic states, free-electron and tight-binding models, semiconductor physics.
Aim:
The aim of the module is to give students a strong grounding in the foundational parts of Quantum Mechanics. This course is basic to an understanding of all modern physics and is prerequisite for all the courses to be followed in the High Energy and Condensed Matter second year Masters programs.
The aim of this module is to introduce students to simple numerical techniques using computers to solve basic problems, which students have never used during their undergraduate studies.
Content:
The first part of this module will introduce students to scientific computing and the role of numerical methods in modeling complicated physical processes on the computer. The structure of this course will involve both lectures and computational hands-on labs to complement the theory learnt in the lectures. Topics, described in detail below, will include numerical recipes (algorithms) for integration, differentiation, differential equations and a gentle introduction to computer programming. The students will learn the basics of FORTRAN programming, which will include hands-on exercises in basic/simple FORTRAN codes. Advanced topics such as numerical linear algebra, Monte Carlo and molecular dynamics will be touched on very briefly in preparation for the second year.
Aim
This module follows the introductory course of Quantum Mechanics I. It aims to continue the development of non-relativistic quantum mechanics as a complete theory of microscopic dynamics, capable of making detailed predictions, with a use of the mathematical methods learnt in previous courses.
Content
The module covers fundamental concepts of quantum mechanics: wave properties, uncertainty principles, Schrödinger equation, and operator and matrix methods. Basic applications of the following are discussed to investigate the rules of quantum mechanics in a more systematic fashion by showing how quantum mechanics is used to examine: the motion of a single particle in one dimension, many particles in one dimension, and a single particle in three dimensions, and angular momentum and spin. The course also examines approximation methods: variational principle and perturbation theory.
Aims:
The course will cover some mathematical techniques commonly used in theoretical physics. This is not a course in pure mathematics, but rather on the application of mathematics to problems of interest in the physical sciences.
Content:
Depending on your initial preparation, to be assessed by a preliminary test that will not count for your final grade, we will cover some or all of the following topics: • Vector calculus in curvilinear coordinates; • The theory of analytic functions; • Linear algebra, vectors and tensors in physics; • Special functions and their physical applications; • Partial and ordinary differential equations; analytical and numerical methods for their solution. This is a course in mathematical physics, so the emphasis will always be on physical applications.
The course is divided into five broad topics, each taking 2-4 weeks or so.
2. LINEAR VECTOR SPACES
1. THEORY OF ANALYTICAL FUNCTION
3. FUNCTION SPACE, ORTHOGONAL POLYNOMIALS, AND FOURIER ANALYSIS
4. DIFFERENTIAL EQUATIONS
5. SPECIAL FUNCTIONS
Details are in the attached course outline (and in the online MSc syllabus here https://eaifr.org/degree-programmes/masters-programme/ or here: https://eaifr.org/media/2825/syllabusmsc.pdf
The aim of the course is to give a complete understanding of Classical Mechanics at an advanced level. The course provides indispensable preparation for all advanced courses in theoretical physics. Techniques learned have wide use in advanced quantum mechanics, classical and quantum field theory, general relativity, particle physics and statistical mechanics.
Content:
1) Lagrangian Formulation of Mechanics: Calculus of Variations, Action Integral, Principle of Least Action, Euler-Lagrange Equation, Generalized Co-ordinates and Momenta, Constraints, Normal Modes, Conservation Laws: energy, linear momentum, angular momentum. Integration of the equations of movement: problems with one degree of freedom, two body problem, movement in a central field, scattering in a central field
2) Hamiltonian Formulation of Mechanics: Hamilton’s Equations, Poisson Brackets, Canonical Transformations, Liouville theorem, adiabatic invariants
3) Continuous Symmetries and Conservations Laws.
The course is in two almost independent parts: Electrodynamics (Part 1) and Special Relativity (Part2)
Part 2 has already been taught by Prof. Gazeau. The syllabus is below. Part I will be taught also.
PART I: ELECTRODYNAMICS
Review of the Coulomb, Gauss’s laws and surface integral, Bio-Savart law, Ampere’s law and line integral, Faraday law. The Divergence Theorem. Conservation of charge and the equation of continuity. Stokes’ Theorem and the meaning of the curl. Simple examples of curl in cylindrical polar coordinates. The displacement current. A more comprehensive study of Lorentz force, Electrostatics, Magnetostatics and Faraday’s law. Maxwell’s equations (non-relativistic form). Conservation laws. Then solving Maxwell’s equations: Retarded solutions, Radiating Systems and Plane waves. This would include boundary problems in electrostatics; the green function; momentum of distributed charges. Electromagnetic waves and their propagation. Generation of electromagnetic waves, Hertz’ experience, qualitative and quantitative transport of electromagnetic waves. Electric and magnetic fields in matter: vector fields E, B, H, D, P and M. Retarded potential. Electrodynamics in relativistic notation. Lorentz transformation for electromagnetic field. Energy and momentum fields. The electromagnetic mass. The dynamics of relativistic particles and fields and the radiation of moving charges once more. Energy transport and the vector electromagnetic energy transport (Poynting Vector).
PART II: SPECIAL RELATIVITY -- Taught already by Prof. GAZEAU
Given the transformation properties of Maxwell’s equations one introduces the concept of Lorentz transformations in general and shows that they preserve the Minkowski metric. Consequences of this are elaborated including: simultaneity of events, length contraction, time dilatation, composition of velocity, transformation of acceleration. These are corroborated by the experiments of Fizeau and Michelson-Morley.
Relativistic properties of particles including the differences between rest mass and relativistic mass are explained, transformations of energy and momentum are given, and relativistic equations of motion the relativistic expression of energy, particle with zero proper mass and conservation laws of energy and momentum are discussed.