Analysis 1

15 credits
Spring teaching, Year 1

Topics 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 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.

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.

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.

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 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.

Mathematical Methods for Physics 2

15 credits
Spring teaching, Year 1

Topics covered include:

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.

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

Optics & Imaging

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.

Physics in Practice

15 credits
Autumn teaching, Year 1

This module covers:

• Dimensions and units
• Estimation of uncertainties: significant figures and decimal places; coupled with practice in 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

Geometry

15 credits
Autumn teaching, Year 1

Topics include: vectors in two and three dimensions. Vector algebra: addition, scalar product, vector product, including triple products. Applications in two- and three-dimensional geometry: points, lines, planes, geometrical theorems. Area and volume. Linear dependence and determinants. Polar co-ordinates in two and three dimensions. Definitions of a group and a field. Polynomials. Complex numbers, Argand plane, De Moivre's theorem. Matrices: addition, multiplication, inverses. Transformations in R^2 and R^3: isometries. Analytical geometry: classification and properties of conics.

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. Non-physics students taking this course should be aware that it includes mathematical content. This is mostly algebraic manipulation of equations, but includes calculus and first-order differential equations. 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.

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 variable; 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. Metric spaces and normed linear spaces: inner products; Cauchy sequences, convergence and completeness; the Euclidean space R^n; introduction to general topology; applications and examples.

Electrodynamics

15 credits
Autumn teaching, Year 2

This module covers 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.

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

Scientific Computing

15 credits
Autumn teaching, Year 2

This module covers the revision of representation of numbers and basics of Python programming. Including 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.

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. 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, to 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

Topics covered include: 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.

BSc Final Year Project

30 credits
Autumn & spring teaching, Year 3

You will undertake individually a two-term 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 second year.

Condensed State Physics

15 credits
Autumn teaching, Year 3

Classification of Solids
1. Types of solids; Classification of elements and compounds by physical properties. 2. Types of bonding. 3. Basic band theory of metals, electrical insulators and semiconductors. 

Crystal Structures 
4. Crystals; Unit cells and lattice parameters. 5. Bravais lattices; Crystallographic basis; Crystal axes and planes. 6. Cubic and hexagonal structures. 7. Reciprocal lattice.

Diffraction by Crystals
8. Physical Processes; Braggs law; Atomic and geometrical scattering factors. 9. Diffraction crystallography. 

Lattice Vibrations
10. Thermal properties of electrical insulators: Specific Heat and Thermal Conductivity. 11. Vibrations of monatomic and diatomic 1-D crystals; Acoustic and Optical modes. 12. Quantisation of Lattice Vibrations; Phonons. 13. Einstein and Debye Models for Lattice Specific Heat.

The Free Electron Model
14. Classical Free Electron Gas. 15. Quantised Free Electron Model. 16. Specific Heat of the Conduction Electrons. 17. Electrical and Thermal Conductivity of metals. 18. AC conductivity and Optical Properties of metals.

Dielectric and Optical Properties of Insulators
19. Dielectric Constant and Polarizability. 20. Sources of Polarizability; 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.

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. Cyclic codes.

Complex Analysis

15 credits
Spring teaching, Year 3

Topics covered 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.

Dynamical Systems

15 credits
Spring teaching, Year 3

Topics covered include:

• General dynamical systems: semiflow, stability and attraction, omega-limit set, global attractor. 
• Ordinary differential equations: Linear systems, Lyapunov function, linearized systems around fixed points, periodic orbits, Two-dimensional systems, centre manifold, bifurcation.
• Discrete systems (iterations): Linear systems, linearized systems around fixed points.

Functional Analysis

15 credits
Spring teaching, Year 3

Topics on this module include: Banach spaces; Banach fixed-point theorem; Baire's Theorem; Bounded linear operators and on Banach spaces; continuous linear functionals; Banach-Steinhaus Uniform Boundedness Principle; open mapping and closed graph theorems; Hahn-Banach Theorem; compactness of sets; Hilbert spaces; orthogonal expansions; Riesz-Fischer Theorem.

Lasers

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.

Medical Statistics

15 credits
Spring teaching, Year 3

Topics include: logistic regression, fitting and interpretation. Survival times; Kaplan-Meier estimate, log-rank test, Cox proportional hazard model. Designing medical research. Clinical trials; phases I-IV, randomised double-blind controlled trial, ethical issues, sample size, early stopping. Observational studies: prospective/retrospective, longitudinal/cross-sectional. Analysis of categorical data; relative risk, odds ratio; McNemar's test, meta-analysis (Mantel-Haenszel method). Diagnostic tests; sensitivity and specificity; receiver operating characteristic. Standardised mortality rates.

Particle Physics

15 credits
Spring teaching, Year 3

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

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

Random processes

15 credits
Spring teaching, Year 3

The aim of this module is to present a systematic introductory account of several principal areas in stochastic processes. You cover basic principles of model building and analysis with applications that are drawn from mainly biology and engineering.

Topics include:

• Poisson processes:
• Definition and assumptions.
• Density and distribution of inter-event time.
• Pooled Poisson process.
• Breaking down a Poisson process.

• Birth processes, birth- and death- processes:
• The simple birth process.
• The pure death process.
• The Kolmogorov equations.
• The simple birth-death process.
• Simple birth-death: extinction.
• An embedded process.
• The immigration-death model.

• Queues:
• The simple M/M/1 queue.
• Queue size.
• The M/M/n queue.
• The M/M/ queue.
• The M/D/1 queue.
• The M/G/1 queue.
• Equilibrium theory.
• Other queues.

• Renewal processes:
• Discrete-time renewal processes.
• The ordinary renewal process.
• The equilibrium renewal process.

• Epidemic models:
• The simple epidemic.
• General epidemic.
• The threshold in epidemic models.