AMA Level 4 Modules
Prerequisite: AMA3002 or, subject to the approval of HoT (AMA), PHY3011.
CoRequisites: None.
Compulsory Element: None.
Lecturer: Dr G De Chiara
Introduction
Quantum mechanics is one of the greatest intellectual achievements of the last century. In this course we will recap the mathematical framework of quantum mechanics, and show how it describes the structure of atoms and their collisions. Atomic, molecular and optical physics is a burgeoning field, with numerous Nobel prizes awarded in the area in the last 20 years. Intense research is ongoing internationally, aimed at obtaining ever more precise understanding of atomic structure and atomic interactions, and how they can be controlled by external fields. Such understanding is key to, e.g., developing new quantumbased technologies for metrology and cryptography, for precision tests of fundamental laws of physics, for the study of astrophysical phenomena, and for many other applications.
Students completing this module will expand the toolkit of mathematical methods of quantum mechanics, and obtain basic theoretical knowledge of atomic structure and introductory manybody quantum physics.
Contents
1). Review of fundamental quantum theory (Postulates of quantum mechanics; Dirac notation; Schrödinger equation; spin1/2 systems; stationary perturbation theory).
2). Coupled angular momenta: spin1/2 coupling; singlet and triplet subspaces for two coupled spin1/2 particles; Coupling of general angular momenta;
3). Spinorbit coupling; fine and hyperfine structures of the hydrogen atom.
4). Timedependent perturbation theory.
5). Elements of collisions and scattering in quantum mechanics.
6). Identical particles and second quantisation; operators representation.
7). Basics of electromagnetic field quantisation.
8). Systems of interacting bosons: BoseEinstein condensation and superfluidity.
Supplementary Notes: None.
Learning Outcomes
On successful completion of the module, it is intended that students will be able to:
1. Use the rules for the construction of a basis for coupled angular momenta.
2. Grasp the fundamental features of the fine and hyperfine structures of the hydrogen atom.
3. Understand the techniques for dealing with timedependent perturbation theory.
4. Apply the theory of scattering to simple quantum mechanical problems.
5. Describe systems of identical particles in quantum mechanics and write the second quantisation representation of operators.
6. Apply the formalism of second quantisation to the electromagnetic field and systems of interacting bosons.
Skills: Mathematical modelling. Problem solving. Abstract thinking.
Assessment
Exam 80%, Group Presentation 20%

AMA4003 Advanced Mathematical Methods (2^{nd} semester)
Prerequisite: While there are no specific prerequisites, 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.
Lecturers: Dr A Brown
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, socalled ‘complete integrals’ of the general firstorder partial differential equation. A complete integral may be viewed as a family of surfaces. Envelopes of this family, and of subfamilies 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 HilbertSchmidt theory of linear integral equations.
While the course is more or less selfcontained, 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 firstorder 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 firstorder 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. HilbertSchmidt Theory: Hermitian kernel  characteristic values and functions; orthonormality. HilbertSchmidt theorem. Solution of Fredholm equation of the second kind.
Assessment
Exam 70% Project 20% Assignment 10%

AMA4004 Statistical Mechanics (2^{nd} semester)
Prerequisite: AMA3002 or PHY3011
Lecturer: Dr G Tribello
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 application we consider are ideal gasses made up of classical and quantum particles, vibrations in solids (phonons) and the Ising model and its variants in the meanfield approximation. We then study the techniques of MonteCarlo and molecular dynamics in order to understand how we can solve statistical mechanics problems approximately. During the course there are multiple opportunities to apply these ideas yourself by writing short pieces of software to solve various problems in statistical mechanics. In addition, the final assessment involves a selfstudy of a modern scientific article on this topic.
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 GibbsDuhem 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: phasespace and partition function. MaxwellBoltzmann distribution. Equipartition and virial theorems.
Applications of Boltzmann statistics to ideal systems: Factorisation approximation. Monoatomic 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. FermiDirac and BoseEinstein statistics: derivation of partition functions. Recovering the classical limit. Statistical mechanics of quasiparticles: phonons and photons. Ideal Fermi gas: low and highdensity limits. Electrons in metals. Ideal Bose gas: low and highdensity limits. BoseEinstein 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 liquidvapor phase transition. Models of Ising and Heisenberg. Exact solution and Mean field theories. Introduction to simulation methods in statistical physics.
Assessment
Exam 45% Project 20% Portfolio 25% Class test 10%

AMA4005 Project (1st and 2nd semester)
Prerequisite: This twosemesterlong double module is only available to students on the Mathematics/Applied Mathematics and Physics/Theoretical Physics MSci pathways.
Coordinator: Prof M Paternostro
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 30minute 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%
Prerequisite: While there are no specific prerequisites, 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.
Lecturer: Dr C Ballance
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 nonrelativistic quantum mechanics for example the problems involve finding numerical solutions to the Schrödinger equation. In fluid mechanics solutions of the NavierStokes 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 BlackScholes 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. Firstorder linear and quasilinear equations. Method of characteristics.
3. Secondorder linear equations and their classification. Canonical forms.
4. The method of separation of variables. Applications.
5. Introduction to numerical methods. Explicit and implicit finitedifference schemes. Applications.
6. Practical assignment. Numerical solutions to model problems.
7. SturmLiouville problem. Eigenfunctions and eigenvalues. Eigenfunction expansion.
8. Elliptic equations.
9. Dirac delta function. Green's functions and integral representation.
Assessment
Exam 70% Project 30%

AMA4009 Information Theory (2nd semester)
Prerequisite: While there are no specific prerequisites, 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.
Lecturer: Prof H van der Hart
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 welldefined 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 (1st semester)
Prerequisite: Knowledge of linear algebra (as in MTH2001 or PMA2020). AMA3002 is NOT required.
Lecturer: Dr A Ferraro, Prof M Paternostro
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 nonlocality, 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.