## Classical Mechanics

**15 credits****Autumn Teaching, Year 1**

An introduction to mechanics and its applications, covering: Newton's Laws; particle dynamics; work and kinetic energy; potential energy and energy conservation; momentum, impulse and collisions; rocket propulsion; rigid-body rotation; torque and angular momentum; gyroscopes and precession; statics and equilibrium; fluid statics and dynamics; gravitation, satellite motion and Kepler's laws.

## Mathematical Methods for Physics 1

**15 credits****Autumn Teaching, Year 1**

Topics covered include:

- Introduction to functions: functions and graphs;
- Classical functions: trigonometry, exponential and logarithmic functions, hyperbolic functions;
- Differentiation: standard derivatives, differentiation of composite functions;
- Curves and functions: stationary points, local/global minima/maxima; graph sketching;
- Integration: standard integrals, integration by parts and substitution, areas, volumes, averages, special integration techniques;
- Power series expansions: Taylor expansions, approximations, hyperbolic and trigonometric functions;
- Convergence of series: absolute convergence; integral test; ratio test
- Complex numbers: complex conjugates, complex plane, polar representation, complex algebra, exponential function, DeMoivre's Theorem;
- Vectors: working with vectors, scalar product of vectors, vector product of vectors;
- Determinants and matrices: definition and properties, matrices and matrix algebra, solutions of systems of linear equations.

The computer lab component of the module will introduce you to Maple.

## Physics in Practice

**15 credits****Autumn Teaching, Year 1**

This module covers:

- Dimensions and units
- Estimation of uncertainties: significant figures and decimal places; reading Vernier scales
- Introduction to spreadsheets
- Mean, standard deviation and standard error: weighted averages, and the uncertainty thereon
- Error propagation: simple formulae covering addition, multiplication and powers; general formula for small error propagation
- Histograms, and manipulation of distributions
- The Gaussian distribution
- Chi squared, and (straight) line fitting
- Identifying and dealing with systematics
- Assessing data quality
- Circuit simulation
- DC circuits: introduction, Ohm's Law, Non-Linear circuit elements, oscilloscopes
- Capacitors: RC circuits, differentiator, integrator, low pass filter, high pass filter

## Analysis 1

**15 credits****Spring Teaching, Year 1**

In this module, the topics you will cover will include:

- Sequences; convergence, Cauchy sequences, subsequences
- Series; proof and application of convergence/divergence criteria
- Limits of functions; definitions, examples and properties
- Continuity; intermediate value theorem, uniform continuity
- Differentiability; definition, proofs of mean value theorems.

## Classical Physics

**15 credits****Spring Teaching, Year 1**

This module is focused around three main areas:

Electromagnetism:

-Electric forces and fields in systems with static discrete electric charges and static observers.

-Continuous charge distributions, Gauss's law. Electric potential energy and electric potential.

-Energy stored by the electric field. Motion of charged particles in static electric fields.

-Conductors and insulators in electric fields. Capacitance and capacitors. Energy storage in capacitors. Dielectrics. Drude's model of conduction.

-Creation of magnetic fields from linear motion of charges (ie, a current) electron spin and orbital motion; motion perpendicular to an electric field. Force on a charged particle moving perpendicular to magnetic field.

Thermodynamics:

-Phases of matter; the zeroth law of thermodynamics; temperature and temperature scales

-Thermal expansion coefficients

-The ideal-gas law

-The kinetic theory of gases; the Maxwell speed distribution; mean free paths; transport properties of gases; the equipartition theorem

-Heat capacity; latent heat

-The first law of thermodynamics; internal energy of gases

-PV diagrams; work

-Adiabatic processes

## Mathematical Methods for Physics 2

**15 credits****Spring Teaching, Year 1**

In this module, you cover:

**Integration of scalar and vector fields**

- surface integrals of functions of two variables using Cartesian and polar coordinates
- surface integrals of functions of three variables using Cartesian, spherical polar and cylindrical polar coordinates
- volume integrals of functions of three variables using Cartesian, spherical polar and cylindrical polar coordinates
- line integrals along two- and three-dimensional curves.

**Differentiation**

- partial differentiation of functions of several variables
- definition and interpretation of partial derivatives
- partial derivatives of first and higher order.

**Differentiation of scalar and vector fields**

- directional derivative
- gradient, divergence and curl and their properties
- theorems of Gauss, Stokes and their applications.

You are also introduced to Python in the computer lab component of this module.

## Oscillations, Waves and Optics

**15 credits****Spring Teaching, Year 1**

This module covers:

- Simple harmonic motion
- Forced oscillations and resonance
- Mechanical waves
- Properties of sound
- Phasors
- Coupled oscillators and normal modes
- Geometrical optics to the level of simple optical systems
- Huygens' principle, introduction to wave optics
- Interference and diffraction at single and multiple apertures
- Dispersion
- Detectors
- Optical cavities and laser action
- Practical introduction to the use of telescopes.

## Foundations of Modern Physics

**15 credits****Autumn Teaching, Year 1**

This module gives you an introduction to the foundations of modern physics.

Topics covered include:

- Historical context, physics in the 19th century
- black-body radiation, Planck's quanta, photons, photoelectric effect, Compton effect
- atoms, atomic spectra, Franck-Hertz and Stern-Gerlach experiments
- Bohr theory of the atom
- De Broglie waves
- Heisenberg uncertainty principle
- bosons and fermions, antimatter, Bose-Einstein condensates
- periodic system of elements
- relativity: Historical perspective
- inertial frames and transformations, Newton's laws in inertial frames
- Michelson-Morley experiment - observed constancy of speed of light, Einstein's assumptions
- Lorentz-Einstein transformations, Minkowski diagrams, Lorentz contraction, time dilation
- transformation of velocities - stellar aberration, variation of mass, mass-energy equivalence
- Lorentz transformations for momentum and energy.

## Introduction to Astrophysics

**15 credits****Autumn Teaching, Year 1**

This module aims to explain the contents, dimensions and history of the Universe, primarily at a descriptive level. It applies basic physical laws to the study of the Universe, enabling simple calculations. This module also includes an introduction to special relativity, which is shared with the Introduction to Modern Physics module. This module covers:

- A brief history of astronomy
- The scale of the universe
- Time and motion in the universe
- Planets, asteroids and comets
- Stars: their birth and death
- The Milky Way and our place within it
- Nebulae
- Galaxies: types, distance, formation, structure
- Cosmology: dynamics of the universe, the Big Bang, the cosmic microwave background
- Relativity: Historical perspective
- Inertial frames and transformations; Newton's laws in inertial frames
- Michelson-Morley experiment – observed constancy of speed of light. Einstein's assumptions
- Lorentz-Einstein transformations; Minkowski diagrams; Lorentz contraction; time dilation
- Transformation of velocities – stellar aberration. Variation of mass, mass-energy equivalence
- Lorentz transformations for momentum and energy.

## Analysis 2

**15 credits****Autumn Teaching, Year 2**

Topics covered:

- Power series, radius of convergence, Taylor series and Taylor's

formula, applications and examples. - Upper and lower sums, the Riemann integral, basic properties of the Riemann

integral, primitive; fundamental theorem of calculus, integration by parts and

change of variabl, applications and examples. - Pointwise and uniform convergence of sequences and series of functions,

interchange of differentiation or integration and limit for sequences and series,

differentiation and integration of power series term by term, applications and

examples.

## Electrodynamics

**15 credits****Autumn Teaching, Year 2**

This is a first module on electro/magnetostatics and electrodynamics in differential form with key applications.

Topics covered include:

- Mathematical Revision
- Electrostatics: equations for the E-field, potential, energy, basic boundary-value setups
- Electrostatics: dielectrics, displacement and free charge
- Magnetostatics: forces, equations for the B-field, vector potential, Biot-Savart, dipole field of current loops
- Magnetostatics: diamagnetism and paramagnetism, auxiliary field H, ferromagnetism
- Electrodynamics: Faraday's law, inductance and back emf, circuit applications, Maxwell-Ampere law, energy and Poynting's theorem
- Electromagnetic waves: wave equation, plane waves, polarization, waves in dielectrics, reflection at an interface, wave velocity/group velocity/dispersion
- Potentials and dipole radiation

## Mathematical Methods for Physics 3

**15 credits****Autumn Teaching, Year 2**

This module teaches mathematical techniques that are of use in physics, in particular relating to the solution of differential equations. It also aims to give experience of mathematical modelling of physical problems. The module includes:

- Fourier series
- Ordinary differential equations
- Some linear algebra
- Fourier and Laplace transform
- Series solutions of differential equations
- Partial differential equations.

## Scientific Computing

**15 credits****Autumn Teaching, Year 2**

This module covers the revision of representation of numbers and basics of Python programming. You also study the application of numerical methods to model simple physical problems, involving:

- solution of algebraic equations
- interpolation
- numerical integration and differentiation
- numerical solution of ordinary differential equations
- numerical solution of linear systems of equations
- visualisation of data.

## Quantum Mechanics 1

**15 credits****Spring Teaching, Year 2**

Module topics include

- Introduction to quantum mechanics, wave functions and the Schroedinger equation in 1D.
- Statistical interpretation of quantum mechanics, probability density, expectation values, normalisation of the wave function.
- Position and momentum, Heisenberg uncertainty relation.
- Time-independent Schroedinger equation, stationary states, eigenstates and eigenvalues.
- Bound states in a potential, infinite square well.
- Completeness and orthogonality of eigenstates.
- Free particle, probability current, wave packets, group and phase velocities, dispersion.
- General potentials, bound and continuum states, continuity of the wave function and its first
- derivative.
- Bound states in a finite square well.
- Left- and right-incident scattering of a finite square well, reflection and transmission probabilities.
- Reflection and transmission at a finite square well.
- Reflection and transmission at a square barrier, over-the-barrier reflection, tunnelling, resonant
- tunnelling through multiple barriers.
- Harmonic oscillator (analytic approach).
- Quantum mechanics in 3D, degeneracy in the 3D isotropic harmonic oscillator.
- Orbital angular momentum, commutators and simultaneous measurement.
- Motion in a central potential, Schroedinger equation in spherical polar coordinates.
- Schroedinger equation in a Coulomb potential.
- H atom.
- Spin, identical particles, spin-statistics theorem.
- Helium, basics of atomic structure.
- Time-independent perturbation theory for non-degenerate bound states.
- Applications of perturbation theory, fine structure in the H atom.
- Schroedinger equation for a particle coupled to an electromagnetic field.
- Summary and revision

## Skills in Physics 2

**15 credits****Spring Teaching, Year 2**

The aims and objectives of the module are to develop and enhance the deployment of a range of skills and knowledge, which should have been acquired in Year 1, to elucidate real problems and/or phenomena. The idea is to improve your abilities to make use of information from appropriate basic modules to solve problems, and to wean you away from the notion that real problems can be solved with the knowledge from a single module.

## Theoretical Physics

**15 credits****Spring Teaching, Year 2**

Topics covered include:

- electrostatics – electrostatic potentials and electric fields, methods of images, Laplace and Poisson equations, introduction to Green's functions, and Gauss' and Stokes' theorems
- magnetostatics – vector potential
- elementary considerations of Function Theory – complex numbers, Cauchy-Riemann differential equations, line-integrals, Cauchy's theorem, Power series, Laurent series, residue theorem, applications in electrostatics
- vector calculus in space-time – four-vectors and tensors, metric tensor, energy-momentum four-vector relativistic electrodynamics, charges seen by different observers, four-vector potential, Maxwell's equations using four-vectors
- calculus of variations, Fermat's principle, Euler-Lagrange equation, definition of action, applications to mechanics and electromagnetism.

## Thermal and Statistical Physics

**15 credits****Spring Teaching, Year 2**

Topics covered include:

- Review of kinetic theory of gases and first law of thermodynamics.
- Basics of statistical mechanics. Microstates, entropy, second law.
- Classical thermodynamics. Engines and refrigerators.
- More statistical mechanics. Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions.
- Blackbody radiation.
- Elements of phase transitions.

## Atomic Physics

**15 credits****Autumn Teaching, Year 3**

This module will cover the following topics:

- Physics of the hydrogen atom
- Relativistic hydrogen atom (fine structure, antimatter)
- Hyperfine structure of hydrogen and the 21-cm line
- Interaction with external fields (Zeeman Effect, Stark Effect)
- Helium atom
- Multi-electron atoms and the periodic system
- Molecules and chemical binding
- Molecular structure: vibration and rotation
- Radiative processes, emission and absorption spectra

## Condensed State Physics

**15 credits****Autumn Teaching, Year 3**

Topics covered on this module include:

**Classification of solids**

- Types of solids; classification of elements and compounds by physical properties
- Types of bonding
- Basic band theory of metals, electrical insulators and semiconductors

**Crystal structures**

- Crystals; unit cells and lattice parameters
- Bravais lattices; crystallographic basis; crystal axes and planes
- Cubic and hexagonal structures
- Reciprocal lattice

**Diffraction by crystals**

- Physical processes; Braggs law; atomic and geometrical scattering factors
- Diffraction crystallography

**Lattice vibrations**

- Thermal properties of electrical insulators: specific heat and thermal conductivity
- Vibrations of monatomic and diatomic 1-D crystals; acoustic and optical modes
- Quantisation of lattice vibrations; phonons
- Einstein and Debye models for lattice specific heat

**The free electron model**

- Classical free electron gas
- Quantised free electron model
- Specific heat of the conduction electrons
- Electrical and thermal conductivity of metals
- AC conductivity and optical properties of metals

**Dielectric and optical properties of insulators**

- Dielectric constant and polarisability
- Sources of polarisability; dipolar dispersion.

## Nuclear and Particle Physics

**15 credits****Autumn Teaching, Year 3**

This module on nuclear and particle physics covers:

- Chronology of discoveries.
- Basic nuclear properties.
- Nuclear forces.
- Models of nuclear structure.
- Magic numbers.
- Nuclear reactions, nuclear decay and radioactivity, including their roles in nature.
- The weak force.
- Existence and properties of neutrinos.
- Qualitative introduction to neutrino oscillations.
- C, P and T symmetries.
- Classification of elementary particles, and their reactions and decays.
- Particle structure.
- Qualitative introduction to Feynman diagrams.

## Mathematical Physics Projects

**30 credits****Autumn & Spring Teaching, Year 3**

You will individually undertake two small theoretical projects under the supervision of a member of teaching staff, generally a different supervisor for each project. The topics will be chosen from a list that will vary from year to year and will be made available to you during the second term of your second year.

## Complex Analysis

**15 credits****Spring Teaching, Year 3**

In this module, the topics you will cover will include:

- holomorphic functions, Cauchy's theorem and its consequences
- power series, integration, differentiation and analysis of convergence
- Taylor expansions and circle of convergence
- Laurent expansions and classification of isolated singularities
- residue theorem and evaluation of integrals
- Rouche's theorem and the fundamental theorem of algebra.

## Quantum Mechanics 2

**15 credits****Spring Teaching, Year 3**

This module on quantum mechanics employing Dirac notation and algebraic methods. Topics covered include:

- Dirac's formulation of quantum mechanics - bras&kets, observables, algebraic treatment of harmonic oscillator, x&p representation, compatibility, uncertainty
- Symmetries and conservation laws - generators of translations&rotations, parity, time evolution, Heisenberg picture
- Angular momentum - algebraic treatment, spin, "addition" of angular momenta, explicit form of rotation operators
- Approximation methods - time-independent perturbation theory: first and second orders, degeneracies; WKB approximation & tunneling
- Interaction picture and time-dependent perturbation theory
- Basics of field quantisation - creation and annihilation operators, EM transitions
- Basic scattering theory
- Mixed states and quantum measurement - density matrix, Bell's inequality
- Elements of relativistic QM and antiparticles

## Advanced Condensed State Physics

**15 credits****Spring Teaching, Year 3**

This module covers the following topics:

- Electronic Energy bands in Solids. Electrons in periodic potentials; Brillouin Zones; Bloch states. Nearly Free Electron (NFE) model. Tight-Binding Approximation (TBA) model. Band structure of selected metals, insulators and semiconductors. Optical Properties.
- Electron Dynamics. Electrons and holes. Effective Mass. Mobilities. Magneto-transport.
- Semiconductors. Classification; Energy Gaps. Donor and Acceptor doping. Equilibrium carrier statistics in intrinsic and doped materials. Temperature dependence of electrical and optical properties.
- Semiconductor Devices. p-n junctions. Diodes, LEDs, Lasers, Transistors. Superlattices and 2DEG devices.
- Lattice Defects. Types of defects. Electronic and optical effects of defects in semiconductors and insulators.

## Coding Theory

**15 credits****Spring Teaching, Year 3**

Topics include:

- introduction to error-correcting codes
- the main coding theory problem
- finite fields
- vector spaces over finite fields
- linear codes
- encoding and decoding with a linear code
- the dual code and the parity check matrix
- Hamming codes
- constructions of codes
- weight enumerators
- MDS codes.

## Dynamical Systems

**15 credits****Spring Teaching, Year 3**

General dynamical systems:

- semiflow
- stability and attraction
- omega-limit set
- global attractor.

Ordinary Differential Equations:

- linear systems
- Lyapunov function
- linearised systems around fixed points
- two-dimensional systems
- periodic orbit.

Discrete systems (iterations):

- linear systems
- linearised systems around fixed points
- chaos.

## Extragalactic Astronomy

**15 credits****Spring Teaching, Year 3**

This module covers:

- Overview of observational cosmology – content of the Universe, incl. current evidence for Dark Matter and Dark Energy; evolution and eventual fate of the Universe; cosmic microwave background radiation; nucleosynthesis
- Galaxy formation – linear perturbation theory; growth and collapse of spherical perturbations; hierarchical galaxy formation models
- Galaxy structure and global properties – morphology; stellar populations; spectral energy distributions; galaxy scaling laws
- Global properties of the interstellar medium
- Statistical properties of the galaxy population – luminosity function; mass function; star-formation history of the Universe
- How to detect astrophysical processes in distant galaxies using modern telescopes
- Black holes and active galactic nuclei
- Galaxy clusters and the intracluster medium; galaxy groups

## Lasers and Photonics

**15 credits****Spring Teaching, Year 3**

This module covers:

- Light-matter interaction.
- Rate equations of lasers.
- Principles of Gaussian optics and optical cavities.
- Types of lasers and their applications.

## Optimal Control of Partial Differential Equations

**15 credits****Spring Teaching, Year 3**

You will be introduced to optimal control problems for partial differential equations. Starting from basic concepts in finite dimensions (existence, optimality conditions, adjoint, Lagrange functional and KKT system) you will study the theory of linear-quadratic elliptic optimal control problems (weak solutions, existence of optimal controls, adjoint operators, necessary optimality conditions, Langrange functional and adjoint as Langrangian multiplier) as well as basic numerical methods for your solution (gradient method, projected gradient method and active set strategy). The extension to semi-linear elliptic control problems will also be considered.

## Particle Physics

**15 credits****Spring Teaching, Year 3**

This module is a first introduction to basic concepts in Elementary Particle Physics. It presents an introductory discussion of leptons and quarks and their interactions in the standard model. Particular emphasis will be given to experimental methodologies and experimental results. A selection of topics covered in this course include:

- Cross-sections and decay rates
- Relativistic kinematics
- Detectors and accelerators
- Leptons
- Quarks and hadrons
- Space-time symmetries
- The quark model
- Electromagnetic interactions
- Strong interactions: QCD, jets and gluons
- Weak interactions and electro-weak unification
- Discrete symmetries
- Aselection of topics in physics beyond the standard model

## Perturbation theory and calculus of variations

**15 credits****Spring Teaching, Year 3**

The aim of this module is to introduce you to a variety of techniques primarily involving ordinary differential equations, that have applications in various branches of applied mathematics. No particular application is emphasised.

Topics covered include –

Dimensional analysis and scaling:

- physical quantities and their measurement
- dimensions
- change of units
- physical laws
- Buckingham Pi Theorem
- scaling.

Regular perturbation methods:

- direct method applied to algebraic equations and initial value problems (IVP)
- Poincar method for periodic solutions
- validity of approximations.

Singular perturbation methods:

- finding approximate solutions to algebraic solutions
- finding approximate solutions to boundary value problems (BVP) including boundary layers and matching.

Calculus of Variations:

- necessary conditions for a function to be an extremal of a fixed or free end point problem involving a functional of integral form
- isoperimetric problems.

## Physics Methods in Finance

**15 credits****Spring Teaching, Year 3**

The module will cover topics including:

- Efficient market hypothesis
- Random walk
- Levy stochastic processes and limit theorems
- Scales in financial data
- Stationarity and time correlation
- Time correlation in financial time series
- Stochastic models of price dynamics
- Scaling and its breakdown
- ARCH and GARCH processes
- Financial markets and turbulence
- Correlation and anti-correlation between stocks
- Taxonomy of a stock portfolio
- Options in idealised markets (to include Black & Scholes formula)
- Options in real markets

## Random processes

**15 credits****Spring Teaching, Year 3**

Topics covered on this module include:

- Rationalisation:
- After the introduction of the Poisson process, birth and death processes as well as epidemics models can be presented in full generality as applications of the pooled Poisson process. At the same time, you will be introduced to the Kolmogorov equations and to the techniques for solving them. Renewal theory is needed to better understand queues, and, for this reason, it is discussed before queues.

- Modernisation:
- A modern introductory course on stochastic processes must include at least a section on compound renewal processes (with a focus on the compound Poisson process) as well as a chapter on the Wiener process and on Ito stochastic calculus. This is necessary given the importance this process has in several applications from finance to physics. Modernisation is achieved by including a new introductory chapter divided into three parts.
- Poisson processes:

Density and distribution of into-event time:

- pooled Poisson process
- breaking down a Poisson process
- applications of the Poisson process, e.g. birth-and-death processes, the Kolmogorov equations

Renewal processes:

- the ordinary renewal process
- the equilibrium renewal process
- the compound renewal process
- applications of renewal processes, queues.

Wiener processes:

- definition and properties
- introduction to stochastic integrals
- introduction to stochastic differential equations.

## Researching STEM

**15 credits****Spring Teaching, Year 3**

This module aims to develop the skills and understanding required for explaining scientific concepts to a range of audiences, with a particular focus on school aged pupils (11-18). At the start of the module there will be a lecture and a seminar to introduce the project. Tutorials will follow to aid the student with their research project.

The module will involve time in an educational institution carrying out a minor research project based on an aspect of science communication. You will consider an area of interest, relating to communicating STEM, and carry out some literature-based research. You will then design and carry out a small research project. This is likely to be based on a small number of class observations in a chosen school, possibly with some intervention activities that you would evaluate based on your literature review.

Assessment is in the form of a written assignment, to include a literature review and evaluations of observations (from the educational institution) based on theory.

## MPhys Final Year Project

**45 credits****Autumn & Spring Teaching, Year 4**

You will undertake individually a significant two-term research project under the supervision of a member of teaching staff. The topic will be chosen from a list that will vary from year to year and will be made available to you during the second term of your third year.

## Advanced Numerical Analysis

**15 credits****Autumn Teaching, Year 4**

This module will cover topics including:

- Iterative methods for linear systems: Jacobi and Gauss-Seidel, conjugate gradient, GMRES and Krylov methods
- Iterative methods for nonlinear systems: fixed point iteration, Newton's method and Inexact Newton
- Optimisation: simplex methods, descent methods, convex optimisation and non-convenx optimisation
- Eigenvalue problems: power method, Von Mises method, Jacobi iteration and special matrices
- Numerical methods for ordinary differential equations: existence of solutions for ODE's, Euler's method, Lindelöf-Picard method, continuous dependence and stability of ODE's
- Basic methods: forward and backward Euler, stability, convergence, midpoint and trapezoidal methods (order of convergence, truncation error, stability convergence, absolute stability and A-stability)
- Runge-Kutta methods: one step methods, predictor-corrector methods, explicit RK2 and RK4 as basic examples, and general theory of RK methods such as truncation, consitency, stability and convergence
- Linear multistep methods: multistep methods, truncation, consistency, stability, convergence, difference equaitons, Dahlquist's barriers, Adams family and backward difference formulas
- Boundary value problems in 1d, shooting methods, finite difference methods, convergence analysis, Galerkin methods and convergence analysis

## Atom Light Interactions

**15 credits****Autumn Teaching, Year 4**

The module deals with the interaction of atoms with electromagnetic radiation. Starting from the classical Lorentz model, the relevant physical processes are discussed systematically. This includes the interaction of classical radiation with two-level atoms and the full quantum model of atom light interactions. Applications such as light forces on atoms and lasers are explored.

## Data Analysis Techniques

**15 credits****Autumn Teaching, Year 4**

This module introduces you to the mathematical and statistical techniques used to analyse data. The module is fairly rigorous, and is aimed at students who have, or anticipate having, research data to analyse in a thorough and unbiased way.

Topics include: probability distributions; error propagation; maximum likelihood method and linear least squares fitting; chi-squared testing; subjective probability and Bayes' theorem; monte Carlo techniques; and non-linear least squares fitting.

## Financial Mathematics

**15 credits****Autumn Teaching, Year 4**

You will study generalized cash flows, time value of money, real and money interest rates, compound interest functions, quations of value, loan repayment schemes, investment project evaluation and comparison, bonds, term structure of interest rates, some simple stochastic interest rate models and project writing.

## Further Quantum Mechanics

**15 credits****Autumn Teaching, Year 4**

Topics covered include:

- Review of 4-vector notation and Maxwell equations.
- Relativistic quantum mechanics: Klein-Gordon equation and antiparticles.
- Time-dependent perturbation theory. Application to scattering processes and calculation of cross-sections. Feynman diagrams.
- Spin-1/2 particles and the Dirac equation. Simple fermionic scatterings.

## General Relativity

**15 credits****Autumn Teaching, Year 4**

This module provides an introduction to the general theory of relativity, including:

- Brief review of special relativity
- Scalars, vectors and tensors
- Principles of equivalence and covariance
- Space-time curvature
- The concept of space-time and its metric
- Tensors and curved space-time; covariant differentiation
- The energy-momentum tensor
- Einstein's equations
- The Schwarzschild solution and black holes
- Tests of general relativity
- Weak field gravity and gravitational waves
- Relativity in cosmology and astrophysics.

## Introduction to Cosmology

**15 credits****Autumn Teaching, Year 4**

This module covers:

- Observational Overview: in visible light and other wavebands; the cosmological principle; the expansion of the universe; particles in the universe.
- Newtonian Gravity: the Friedmann equation; the fluid equation; the aceleration equation.
- Geometry: flat, spherical and hyperbolic; infinite vs. observable universes; introduction to topology
- Cosmological Models: solving equations for matter and radiation dominated expansions and for mixtures (assuming flat geometry and zero cosmological constant); variation of particle number density with scale factor; variation of scale factor with time and geometry.
- Observational Parameters: Hubble, density, deceleration.
- Cosmological Constant: fluid description; models with a cosmological constant.
- The Age of the Universe: tests; model dependence; consequences
- Dark Matter: observational evidence; properties; potential candidates (including MACHOS, neutrinos and WIMPS)
- The Cosmic Microwave Background: properties; derivation of photo to baryon ratio; origin of CMB (including decoupling and recombination).
- The Early Universe: the epoch of matter-radiation equality; the relation between temperature and time; an overview of physical properties and particle behaviour.
- Nucleosynthesis: basics of light element formation; derivation of percentage, by mass, of Helium; introduction to observational tests; contrasting decoupling and nucleosynthesis.
- Inflation: definition; three problems (what they are and how they can be solved); estimation of expansion during Inflation; contrasting early time and current inflationary epochs; introduction to cosmological constant problem and quintessence.
- Initial Singularity: definition and implications.
- Connection to General Relativity: brief introduction to Einstein equations and their relation to the Friedmann equation.
- Cosmological Distance Scales: proper, luminosity, angular distances; connection to observables.
- Structures in the Universe: CMB anisotropies; galaxy clustering
- Constraining Cosmology: connection to CMB, large scale structure (inc BAO and weak lensing) and supernovae.

## Mathematical Fluid Mech

**15 credits****Autumn Teaching, Year 4**

The aim of this module is to provide an introduction to fluid mechanics, regarded from the perspective of the mathematical analysis of underlying PDE models. As such the course is at the interface between pure and applied mathematics.

The mdoule focuses on the basic equations of fluid dynamics, namely the Navier-Stokes and Euler equations. These are the equations governing the motion of fluids, such as water or air.

The module starts with the derivation of the basic conservation laws. Some simple cases of solutions are analyzed in detail and then a general existence theory in bounded and unbounded domain is obtained, based on energy methods.

## Programming in C++

**15 credits****Autumn Teaching, Year 4**

After a review of the basic concepts of the C++ language, you are introduced to object oriented programming in C++ and its application to scientific computing. This includes writing and using classes and templates, operator overloading, inheritance, exceptions and error handling. In addition, Eigen, a powerful library for linear algebra is introduced. The results of programs are displayed using the graphics interface dislin.

## Quantum Field Theory 1

**15 credits****Autumn Teaching, Year 4**

This module is an introduction into quantum field theory, covering:

- Action principle and Lagrangean formulation of mechanics
- Lagrangean formulation of field theory and relativistic invariance
- Symmetry, invariance and Noether's theorem
- Canonical quantization of the scalar field
- Canonical quantization of the electromagnetic field
- Canonical quantization of the Dirac spinor field
- Interactions, the S matrix, and perturbative expansions
- Feynman rules and radiative corrections.

## Quantum Optics and Quantum Information

**15 credits****Autumn Teaching, Year 4**

The module will introduce you to quantum optics and quantum information, covering:

- Quantum systems and the qubit
- Non-locality in quantum mechanics
- Methods of quantum optics
- The density matrix
- The process of measurement
- Introduction of irreversibility
- Decoherence and quantum information
- Quantum and classical communication
- Measures of entanglement and distance between states
- Logic operations and quantum algorithms
- Requirements for quantum computers
- Physical systems for quantum information processing.

## Symmetry in Particle Physics

**15 credits****Autumn Teaching, Year 4**

The module provides an introduction into group theory and aspects of symmetry in particle physics, covering:

- Groups and representations
- Lie groups and Lie algebras
- Space-time symmetries and Poincare group
- Symmetry and conservation laws
- Global, local, and discrete symmetry
- Symmetry breaking and the origin of mass
- Symmetry of the standard model, CKM matrix, neutrino masses, tree-level interactions.

## Advanced Partial Differential Equations

**15 credits****Spring Teaching, Year 4**

You will be introduced to modern theory of linear and nonlinear Partial Differential Equations. Starting from the theory of Sobolev spaces and relevant concepts in linear operator theory, which provides the functional analytic framework, you will treat the linear second-order elliptic, parabolic, and hyperbolic equations (Lax-Milgram theorem, existence of weak solutions, regularity, maximum principles), e.g., the potential, diffusion, and wave equations that arise in inhomogeneous media.

The emphasis will be on the solvability of equations with different initial/boundary conditions, as well as the general qualitative properties of their solutions. They then turn to the study of nonlinear PDE, focusing on calculus of variation.

## Advanced Particle Physics

**15 credits****Spring Teaching, Year 4**

You will acquire an overview of the current status of modern particle physics and current experimental techniques used in an attempt to answer today's fundamental questions in this field.

The topics discussed will be:

- Essential skills for the experimental particle physicist
- Neutrino physics: Neutrino oscillations and reactor neutrinos
- Neutrino physics: SuperNova, geo- and solar- neutrinos and direct neutrino mass measurements
- Cosmic ray physics
- Dark matter
- Introduction to QCD (jets, particles distribution functions, etc)
- Higgs physics
- BSM (including supersymmetry)
- Flavour physics & CP violation
- Electric dipole measurements
- Future particle physics experiments.

## Astrophysical Processes

**15 credits****Spring Teaching, Year 4**

This module covers:

- Basic properties of interstellar medium and intergalactic medium
- Radiative transfer
- Emission and absorption lines, line shapes
- Hyperfine transitions, 21-cm line of hydrogen
- Gunn-Peterson effect, Lyman-alpha forest, Damped Lyman Alpha systems
- Radiative heating and cooling processes
- Compton heating/cooling, Sunyaev-Zeldovich effect
- Emission by accelerating changes, retarded potentials, thermal bremstrahlung
- Applications of Special Relativity in Astrophysics, relativistic beaming
- Plasma effects, Faraday rotation, Synchrotron emission
- HII regions, re-ionization

**Module outline**

Specific aims are to provide you with:

- An overview of instrumentation and detectors
- An overview of some of the topical cutting edge questions in the field.

An appreciation of how scientific requirements translate to instrument/detector requirements and design.

- A crash course in Astronomy & Astrophysics (6 hours and directed reading)
- Fluxes, luminosities, magnitudes, etc.
- Radiation processes, black bodies, spectra
- Stars
- Galaxies
- Planets
- Cosmology
- Key questions
- Key requirements
- Telescopes & Instruments (3 hours student-led seminars from reading)
- Optical telescopes
- Interferometry
- Cameras
- Spectroscopy
- Astronomy beyond the e/m spectrum
- Detectors by wavelength (6 hours taught and 3 hours seminars)
- Gamma
- X-ray
- UV
- Optical
- NIR
- Mid-IR
- FIR
- Sub-mm
- Radio
- Detector selection for a future space mission X (4 x 3 hours)
- Scientific motivation and requirements
- Detector options
- External Constraints, financial, risk, etc.
- Detector selection

**Learning Outcomes**

By the end of the courses, you should be able to:

- Display a basic understanding of detectors in astronomy
- Display communication skills
- Distil technological requirements from scientific drivers
- Make an informed choice of detector for given application with justification.

## Beyond the Standard Model

**15 credits****Spring Teaching, Year 4**

This module covers:

- Basics of global supersymmetry: motivation and algebra, the Wess-Zumino model, superfields and superspace, construction of supersymmetry-invariant Lagrangians.
- Weak scale supersymmetry: the gauge hierarchy problem, the Minimal Supersymmetric Standard Model (MSSM).
- Grand unification: SUS(5), the gauge sector, fermion masses, proton decay.
- Extra dimensions: Kaluza-Klein reduction for scalars, fermions and gauge fields, generation of hierarchies, warped geometry.

## Differential Geometry

**15 credits****Spring Teaching, Year 4**

On this module, we will cover:

- Manifolds and differentiable structures
- Lie derivatives
- Parallel transport
- Riemannian metrics and affine connections
- Curvature tensor
- Sectional curvature
- Scalar curvature
- Ricci curvature
- Bianchi identities
- Schur's lemma
- Complete manifolds
- Hopf-Rinow theorem
- Hadmard's theorem
- Geodescis and Jacobi fields
- Bonnet-Meyer and Synge theorems
- Laplace-Beltrami operator
- Heat kernels and index theorem.

## Early Universe

**15 credits****Spring Teaching, Year 4**

An advanced module on cosmology.

Topics include:

- Hot big bang and the FRW model; Redshifts, distances, Hubble law
- Thermal history, decoupling, recombination, nucleosynthesis
- Problems with the hot big bang and inflation with a single scalar field
- Linear cosmological perturbation theory
- Quantum generation of perturbations in inflation
- Scalar and tensor power spectrum predictions from inflation
- Perturbation evolution and growth after reheating; free streaming and Silk damping
- Matter power spectrum and CMB anisotropies.

## Financial Portfolio Analysis

**15 credits****Spring Teaching, Year 4**

You will study valuation, options, asset pricing models, the Black-Scholes model, Hedging and related MatLab programming. These topics form the most essential knowledge for you if you intend to start working in the financial fields. They are complex application problems. Your understanding of mathematics should be good enough to understand the modelling and reasoning skills required. The programming element of this module makes complicated computations manageable and presentable.

## Introduction to Nano-materials and Nano-characterisation

**15 credits****Spring Teaching, Year 4**

Learn the most important analytical techniques used in the nano-physics laboratory today and discuss some of their applications in Materials Physics and nanotechnology where designing devices and functionality at the molecular scale is now possible.

In this module, you cover:

- the basic physical mechanisms of the interaction between solid matter and electromagnetic radiation, electrons and ions
- the principles and usage of microprobes, electron spectroscopy techniques (AES and XPS), x-ray diffraction, electron microscopy (SEM and TEM), light optical microscopy, atomic force microscopy (AFM), scanning tunneling microscopy, Raman spectroscopy and time-resolved optical spectroscopy.

The module includes a coursework component. This involve preparing and giving a presentation on a selected advanced topic related to recent breakthroughs in nanophysics.

Each group will carry out an extensive literature review on a given topic and subsequently prepare and present a 30-minute presentation on their findings.

In your presentation, you are expected to highlight the usefulness of advanced analytical techniques used by researchers in the given subject area.

## Measure and Integration

**15 credits****Spring Teaching, Year 4**

In this module, you cover:

- countably additive measures, sigma-algebras, Borel sets, measure spaces
- outer measures and Caratheodory's construction of measures
- construction and properties of Lebesgue measure in Euclidean spaces
- measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties
- convergence theorems: monotone convergence, dominated convergence, Fatou's lemma
- application of limit theorems to continuity and differentiability of integrals depending on a parameter
- properties of finite measure spaces and probability theory.

## Monte Carlo Simulations

**15 credits****Spring Teaching, Year 4**

The module will cover topics including:

- Introduction to R
- Pseudo-random number generation
- Generation of random variates
- Variance reduction
- Markov-chain Monte Carlo and its foundations
- How to analyse Monte Carlo simulations
- Application to physics: the Ising model
- Application to statistics: goodness-of-fit tests