School of Mathematics and Physics

AMA Level 4 Modules

  • AMA4001 Advanced Quantum Theory (1st semester)

Pre-requisite: AMA3002 or, subject to the approval of HoT  PHY3011

Introduction
The course is intended as a follow on from AMA3002 Quantum Theory.

Contents

1) Review of fundamental quantum theory (Postulates of quantum mechanics; Dirac notation; Schrödinger equation; spin-1/2 systems; stationary perturbation theory).
2) Particle in a central potential. Hydrogen atom.
3) Coupled angular momenta: spin-1/2 coupling; singlet and triplet subspaces for two coupled spin-1/2 particles; Coupling of general angular momenta;
4) Spin-orbit coupling and the fine structure of the hydrogen atom.
5)  Time-dependent perturbation theory.
6) Elements of collisions and scattering in quantum mechanics.
7) Identical particles and second quantisation; operators representation.
8) Basics of electromagnetic field quantisation.
9) Systems of interacting bosons.

Assessment

Exam 80%   Assignment 20%

  • AMA4003 Advanced Mathematical Methods (2nd semester)

Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 4 of an MSci Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

Introduction
This course falls into two equal parts. The first part is devoted to a study of partial differential equations of the first order. Its orientation is very much geometrical. Here the solution of a partial differential equation is envisaged as a surface in the space of the dependent and independent variables. It is first shown how solution surfaces of the quasilinear Lagrange equation may be generated from families of curves. Knowledge of how to solve this equation enables us to find, by methods due to Charpit and Jacobi, so-called ‘complete integrals’ of the general first-order partial differential equation. A complete integral may be viewed as a family of surfaces. Envelopes of this family, and of sub-families of this family, also solve the differential equation and give solutions known as ‘singular integrals’ and ‘general integrals’.

The second half of the course deals with linear integral equations. Often many problems can be formulated in terms of an integral equation rather than the more familiar differential equation. This can have some advantages, for example, usually the boundary conditions are automatically included in the integral equation formulation and do not have to be imposed afterwards as is usual with differential equations. In addition, linear integral equation theory represents a generalization of matrix theory in which the discrete indices of a matrix element become continuous variables.

There are many analogies between matrix theory and linear integral equation theory. For example, a linear matrix equation is solved by constructing the inverse matrix. Similarly, a solution of a linear integral equation is obtained if we can find the resolvent kernel. In matrix theory Hermitian matrices, with their eigenvalues and eigenvectors, play an important role. Analogously, Hermitian kernels, with their characteristic values and characteristic functions, play an important part in the Hilbert-Schmidt theory of linear integral equations.

While the course is more or less self-contained, some basic knowledge of other areas of mathematics is required, in particular, competence at solving ordinary differential equations, a clear understanding of partial differentiation, matrix theory, and, for the section on integral transform methods for solving integral equations, some knowledge of complex contour integration (complex variable theory).

Contents
Partial Differential Equations of the First Order: Integral surface. Curves parallel to the vector (P,Q,R). Families of curves. Surfaces generated by curves. Envelope of a family of curves. Surfaces: normals, envelopes. Lagrange's equation: a solution; the most general solution; integral surface passing through a given curve; characteristic curves; generalization to n independent variables.
The general first-order equation: classification of integrals, complete integral, singular integral, general integral; integral surface passing through a given curve; Charpit's method for finding a complete integral; Cauchy's method of characteristics; characteristic strips; special types of first-order equations, equations only involving the derivatives, equations not involving the independent variables, separable equations, Clairaut equations; Jacobi's method for finding a complete integral, generalization to n independent variables.
Integral Equations: Fredholm and Volterra equations of the first and second kinds; kernel. Operator notation and scalar product. Solving integral equations : general solution; characteristic values and characteristic functions; resolvent kernel; separable kernel; conversion to differential equation. Connection with matrix theory. Square integrable functions and kernels. Singular kernel. Singular integral equation.
Use of Integral Transforms to Solve Integral Equations: Convolution kernels. Laplace and Fourier Transforms; convolution theorems. Application to Fredholm and Volterra equations with convolution kernels. Abel's integral equation.
Solution of Integral Equations by the Method of Successive Approximations: Neumann series; iterated kernels; resolvent kernel; convergence properties of Neumann series.
Degenerate Kernels: Fredholm Formulae for Continuous Kernels; Fredholm determinant; recurrence relations; characteristic values. Hilbert-Schmidt Theory: Hermitian kernel - characteristic values and functions; orthonormality. Hilbert-Schmidt theorem. Solution of Fredholm equation of the second kind.

Assessment

Exam 70%   Project 20%   Assignment 10%

  • AMA4004 Statistical Mechanics (2nd semester)

Pre-requisite: AMA3002 or PHY3011

Introduction
Statistical mechanics is a formalism that aims at explaining the physical properties of matter at the macroscopic level in terms of the dynamical behavior of its microscopic constituents. The scope of the formalism is almost unlimited as it is applicable to matter in any state of aggregation, ranging from gases, liquids and solids to matter in equilibrium with radiation and biological specimens. The aim of this course is to introduce the basic principles and methods of statistical mechanics and to apply them to a number of model systems in order to illustrate their use and potential in a systematic manner.

We start by introducing the fundamentals of thermodynamics and then proceed to develop the concepts and techniques needed to evaluate probability distributions and partition functions. This later quantity establishes the link between the microscopic description of a system, based on quantum states or positions in phase space, and the macroscopic characterization provided by a small set of independent thermodynamic variables.

Among the applications we consider ideal gases made up of classical and quantum particles, vibrations in solids (phonons) and electromagnetic radiation quanta (photons), the behavior of electrons in metals and Bose-Einstein condensation. We then move to interacting classical systems, where we consider the treatment of real gases and liquids and introduce the concept of phase transition. The Ising model and some of its variants are solved in the mean-field approximation. Time permitting, we provide a general introduction to Monte Carlo methods, a set of powerful numerical techniques used to investigate many-body interacting systems.

Contents
Fundamentals of thermodynamics: Systems, phases and state quantities. Equilibrium and temperature. Equations of state. Reversible and irreversible processes. Work and heat. The laws of thermodynamics. Entropy and the second law. Global and local equilibrium. Homogeneous functions: Euler's equation and Gibbs-Duhem relations. Thermodynamic potentials. The principle of maximum entropy. Entropy and energy as thermodynamic potentials. Legendre transformations. Maxwell relations. Jacobi transformations. Phase and chemical equilibrium. Response functions. Thermodynamic stability.
Equilibrium statistical mechanics: Quantum states and phase space. Ensemble theory. The microcanonical ensemble. Connection with thermodynamics: density of states and entropy. The canonical ensemble. Canonical partition function and free energy. Internal energy and energy fluctuations. Microscopic description of heat and mechanical work. The grand canonical ensemble. Grand canonical partition function and grand potential. Density and energy fluctuations. Entropy maximisation: a general method to derive distribution functions. Fluctuations and response functions. Equivalence between ensembles. Final considerations on Boltzmann statistics. Classical statistical mechanics: phase-space and partition function. Maxwell-Boltzmann distribution. Equipartition and virial theorems.
Applications of Boltzmann statistics to ideal systems: Factorisation approximation. Mono-atomic gases. Gases with internal degrees of freedom: vibrations and rotations. Chemical equilibrium and Saha's ionisation formula.
Quantum statistical mechanics: Indistinguishable quantum particles and symmetry requirements. Fermi-Dirac and Bose-Einstein statistics: derivation of partition functions. Recovering the classical limit. Statistical mechanics of quasi-particles: phonons and photons. Ideal Fermi gas: low and high-density limits. Electrons in metals. Ideal Bose gas: low and high-density limits. Bose-Einstein condensation.
Statistical mechanics of interacting systems: Interaction potentials. Perturbation theory using a control parameter. Cluster and virial expansions. The van der Waals equation of state and the liquid-vapor phase transition. Models of Ising and Heisenberg. Exact solution and Mean field theories. Introduction to simulation methods in statistical physics.

Assessment

Exam 50%   Presentation & Report 20%   Logbook 20%   Class test 10%

  • AMA4005 Project (full year)

Pre-requisite: This two-semester-long double module is only available to students on the Mathematics/Applied Mathematics and Physics/Theoretical Physics MSci pathways.

Introduction
The project involves a substantial investigation of a research problem incorporating literature survey, development of appropriate theoretical models and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed. Each student will work under individual supervision of a member of staff.

Contents
The mathematical contents of the project will depend on the nature of the research problem.

Assessment

Dissertation 80%   Presentation 20%

  • AMA4006 Practical Methods for Partial Differential Equations (1st semester)

Pre-requisite: While there are no specific pre-requisites, this course is intended for students at stage 4 of an MSci Mathematics/Applied Mathematics and Physics/Theoretical Physics pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

Introduction
Many problems in applied mathematics reduce to solving either ordinary or, more usually, partial differential equations subject to certain boundary conditions. In many cases of practical interest exact analytical solutions are not available.

In non-relativistic quantum mechanics for example the problems involve finding numerical solutions to the Schrödinger equation. In fluid mechanics solutions of the Navier-Stokes equation are needed in the context for example of weather systems, or of hypersonic flow around a space shuttle, or the flow of blood through arteries. In financial mathematics solutions of the Black-Scholes equation that describes the pricing of options are required. A familiar partial differential equation arising in many problems in physics and engineering is the diffusion equation, which describes how heat flows from hot to cold regions and many other processes. Another familiar example is the wave equation that governs phenomena all around us: electromagnetic waves, the vibrations of guitar strings, and the propagation of the sound they produce.

In order to solve these or other problems we need to understand first the conditions that give rise to a unique solution. Since analytical solutions are only rarely available we examine techniques that have been developed to acquire a numerical approximation to the solution of a particular problem. The third feature of the course is to actually find numerical solutions to a set of representative model problems, by writing computer programs to implement the requisite methods.

Contents
1. Introduction. Examples of partial differential equations in mathematical physics.
2. First-order linear and quasi-linear equations. Method of characteristics.
3. Second-order linear equations and their classification. Canonical forms.
4. The method of separation of variables. Applications.
5. Introduction to numerical methods. Explicit and implicit finite-difference schemes. Applications.
6. Practical assignment. Numerical solutions to model problems.
7. Sturm-Liouville problem. Eigenfunctions and eigenvalues. Eigenfunction expansion.
8. Elliptic equations.
9. Dirac delta function. Green's functions and integral representation.

Assessment

Exam 50%   3x Projects 50% (20%/15%/15%)

  • AMA4009 Information Theory (This module has now moved to 2nd semester)

Pre-requisite: While there are no specific pre-requisites, this course is intended primarily for students at stage 4 of the MSci Mathematics/MSOR/M&CS pathways, and a mathematical knowledge and ability commensurate with this stage is assumed.

Introduction
Information is a message received and understood. It was Claude Shannon who, in 1948, came up with an idea to quantify the amount of information with a well-defined means of mathematics. Shannon's creation of the subject of information theory was one of the great intellectual achievements of the 20th century and it is an important basis of the development of information technology in the 21st century. Theory of information, which may come in various forms including a sound pattern, a page of a book and a TV message, is one of the important applications of mathematics for the modern society. In fact, information theory is not merely an application of mathematics. Information theory has had an important and significant influence on mathematics and has given profound contribution to pure mathematics.

Information transfer occurs through "codes". These codes can take various forms, ranging from the English language to the Morse code to the ASCII code for computer data storage. In this module, we will start by defining what we mathematically mean with a code. To measure the amount of information stored within a message expressed in the code, we will then introduce the key quantity of information theory, entropy. We will also introduce the basics of coding theory: if a message is received with errors, how can we retrieve the original information content.

Some of the basic concepts of probability theory are discussed. When the information is transmitted from one point to another, we quantify how much information is transmitted through a noisy communication channel. We discuss how the information content of a random variable can be measured. How to infer the original message from the distorted data received is considered.

Contents
Introduction to information theory. Random variables and some concepts of probabilities. Basic modular arithmetic and factoring. Secret coding. Uniquely decipherable and instantaneous codes. Optimal codes and Huffman coding. Code extensions. Entropy, conditional entropy, joint entropy and mutual information. Shannon noiseless coding theorem. Noisy information channels. Binary symmetric channel. Decision rules. The fundamental theorem of information theory. Basic coding theory. Linear codes.

Assessment

Exam 70%   Report 30%

  • AMA4021 Mathematical Methods for Quantum Information Processing   (This module has now moved to 1st semester)

Pre-requisite: Knowledge of linear algebra as in AMA2003 or PMA2007. AMA3002 is NOT required.

Introduction
We are currently witnessing an information revolution: digital devices are everywhere around us. If the incredible level of miniaturisation of electronic devices continues at the current pace, in a few years the elementary components will be made of a few atoms. At this level, physical effects ruled by quantum mechanics will start playing a major role. Inspired by this change of perspective, quantum information processing has been developed as a new framework for future computers. The logic of these quantum computers is different from the traditional computers, as the elementary unit of information, the quantum bit, can be in a superposition of two states 0 and 1.

The aim of this module is to give a broad overview of this emerging field by introducing the most important applications: quantum computing, quantum communication (including teleportation) and entanglement. Quantum mechanics is not a requisite and will be introduced at the beginning of the module as an abstract framework in linear algebra.

Students completing this module will acquire knowledge of the mathematical concepts of quantum information processing with possible applications in theoretical physics, applied and pure maths and computer science.

Contents
1. Operatorial quantum mechanics: review of linear algebra in Dirac notation; basics of quantum mechanics for pure states. 2. Density matrix and mixed states; Bloch sphere; generalised measurements. 3. Maps and operations: complete positive maps; Kraus operators; master equations. 4. Quantum Communication protocols: quantum cryptography; cloning; teleportation; dense coding. 5. Quantum computing: review of classical circuits and logic gates; quantum circuits and algorithms; physical implementation of quantum computers. 6. Theory of entanglement: entanglement basics; detection of entanglement; measures of entanglement; entanglement and non-locality, Bell's inequality; multipartite entanglement.

Assessment

Exam 70%   Report 30%

Textbooks
1. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press. 2. J. Preskill, Quantum Information and Computation, Lecture Notes for Physics 229, available at: http://theory.caltech.edu/~preskill/ph229/#lecture.