Modules

• Year 1

  Core Modules

  Mathematical Methods 1 (30 credits)

  ×

  Mathematical Methods 1

  Overview

  Review of A-level calculus: elementary functions and their graphs, domains and ranges, trigonometric functions, derivatives and differentials, integration. Maclaurin expansion. Complex numbers and Euler’s formula.
  
  Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.
  
  Vectors in 3D, definitions and notation, operations on vectors, scalar and vector products, triple products, 2x2 and 3x3 determinants, applications to geometry, equations of a plane and straight line. Rotations and linear transformations in 2D, 2x2 and 3x3 matrices, eigenvectors and eigenvalues.
  
  Newtonian mechanics: kinematics, plane polar coordinates, projectile motion, Newton’s laws, momentum, types of forces, simple pendulum, oscillations (harmonic, forced, damped), planetary motion (universal law of gravity, angular momentum, conic sections, Kepler’s problem).
  
  Curves in 3D (length, curvature, torsion). Functions of several variables, derivatives in 2D and 3D, Taylor expansion, total differential, gradient (nabla operator), stationary points for a function of two variables. Vector functions; div, grad and curl operators and vector operator identities. Line integrals, double integrals, Green's theorem. Surfaces (parametric form, 2nd-degree surfaces). Curvilinear coordinates, spherical and cylindrical coordinates, orthogonal curvilinear coordinates, Lame coefficients. Volume and surface integrals, Gauss's theorem, Stokes's theorem. Operators div, grad, curl and Laplacian in orthogonal curvilinear coordinates.

  Learning Outcomes

  On completion of the module, the students are expected to be able to:
  • Sketch graphs of standard and other simple functions;
  • Use of the unit circle to define trigonometric functions and derive their properties;
  • Integrate and differentiate standard and other simple functions;
  • Expand simple functions in Maclaurin series and use them;
  • Perform basic operations with complex numbers, derive and use Euler's formula;
  • Solve first-order linear and variable separable differential equations;
  • Solve second-order linear differential equations with constant coefficients (both homogeneous and inhomogeneous), identify complementary functions and particular integrals, and find solutions satisfying given initial conditions;
  • Perform operations on vectors in 3D, including vector products, and apply vectors to solve a range of geometrical problems; derive and use equations of straight lines and planes in 3D;
  • Calculate 2x2 and 3x3 determinants;
  • Use matrices to describe linear transformations in 2D, including rotations, and find eigenvalues and eigenvectors for 2x2 matrices.
  • Define basis quantities in mechanics, such as velocity, acceleration and momentum, and state Newton’s laws;
  • Use calculus for solving a range of problems in kinematics and dynamics, including projectile motion, oscillations and planetary motion;
  • Define and recognise the equations of conics, in Cartesian and polar coordinates;
  • Investigate curves in 3D, find their length, curvature and tension;
  • Find partial derivatives for a function of several variables;
  • Expand functions of one and two variables in the Taylor series and investigate their stationary points;
  • Find the partial differential operators div, grad and curl for scalar and vector fields;
  • Calculate line integrals along curves;
  • Calculate double and triple integrals, including surface and volume integrals;
  • Transform between Cartesian, spherical and cylindrical coordinate systems;
  • Investigate simple surfaces in 3D and evaluate surface for the shapes such as the cube, sphere, hemisphere or cylinder;
  • State and apply Green's theorem, Gauss's divergence theorem, and Stokes's theorem

  Skills

  • Proficiency in calculus and its application to a range of problems.
  • Constructing and clearly presenting mathematical and logical arguments.
  • Mathematical modelling and problem solving.
  • Ability to manipulate precise and intricate ideas.
  • Analytical thinking and logical reasoning.

  Coursework

  15%

  Examination

  85%

  Practical

  0%

  Stage/Level

  1

  Credits

  30

  Module Code

  MTH1021

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Introduction to Algebra and Analysis (30 credits)

  ×

  Introduction to Algebra and Analysis

  Overview

  Elementary logic and set theory, number systems (including integers, rationals, reals and complex numbers), bounds, supremums and infimums, basic combinatorics, functions.
  Sequences of real numbers, the notion of convergence of a sequence, completeness, the Bolzano-Weierstrass theorem, limits of series of non-negative reals and convergence tests.
  Analytical definition of continuity, limits of functions and derivatives in terms of a limit of a function. Properties of continuous and differentiable functions. L'Hopital's rule, Rolle's theorem, mean-value theorem.
  Matrices and systems of simultaneous linear equations, vector spaces, linear dependence, basis, dimension.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able:
  • to understand and to apply the basic of mathematical language;
  • use the language of sets and maps and understand the basic properties of sets (finiteness) and maps (injectivity, surjectivity, bijectivity);
  • demonstrate knowledge of fundamental arithmetical and algebraic properties of the integers (divisibility, prime numbers, gcd, lcm) and of the rationals;
  • Solve combinatorial counting problems in a systematic manner.
  • Understand the fundamental properties of the real numbers (existence of irrational numbers, density of Q, decimal expansion, completeness of R).
  • Understand the notions of a sequence of real numbers, including limits, convergence and divergence.
  • Define convergence of infinite series.
  • Investigate the convergence of infinite series using convergence tests.
  • Define limits of functions and define continuous functions.
  • Prove that a function is continuous or discontinuous.
  • Prove and apply basic properties of continuous functions including the intermediate value theorem and the existence of a maximum and a minimum on a compact interval.
  • Define a differentiable function and a derivative.
  • Prove whether a function is differentiable.
  • Calculate (using analysis techniques) derivatives of many types of functions.
  • Understand, apply and prove Rolle's theorem and the Mean Value Theorem.
  • Prove the rules of differentiation such as the product.
  • Understand and apply the theory of systems of linear equations.
  • Produce and understand the definitions of vector space, subspace, linear independence of vectors, bases of vector spaces, the dimension of a vector space.
  • Apply facts about these notions in particular examples and problems.
  • Understand the relation between systems of linear equations and matrices.

  Skills

  • Understanding of part of the main body of knowledge for mathematics: analysis and linear algebra.
  • Logical reasoning.
  • Understanding logical arguments: identifying the assumptions made and the conclusions drawn.
  • Applying fundamental rules and abstract mathematical results, equation solving and calculations; problem solving.

  Coursework

  0%

  Examination

  100%

  Practical

  0%

  Stage/Level

  1

  Credits

  30

  Module Code

  MTH1011

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Scientific Skills (20 credits)

  ×

  Scientific Skills

  Overview

  Experimental Methods:
  Uncertainties, statistics, safety, using standard instruments
  
  Experimental Investigation:
  Performing experiments on a range of phenomena in Physics, recording observations and results
  
  Writing Skills:
  Scientific writing, writing abstracts, writing reports, writing for a general audience
  
  Oral Communication:
  Preparing and executing oral presentations
  
  Computer Skills:
  Using high level computing packages to analyse and present data, and solve problems computationally

  Learning Outcomes

  Plan, execute and report the results of an experiment, and compare results critically with predictions from theory
  
  Communicate scientific concepts in a clear and concise manner both orally and in written form.
  
  Use mathematical software packages to analyse and present data, and solve problems computationally

  Skills

  Work independently and in collaboration with one or two laboratory partners. Searching for and evaluating information from a range of sources. Writing with an appropriate regard for the needs of the audience. Time management and the ability to meet deadlines.

  Coursework

  70%

  Examination

  0%

  Practical

  30%

  Stage/Level

  1

  Credits

  20

  Module Code

  PHY1004

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Foundation Physics (40 credits)

  ×

  Foundation Physics

  Overview

  Classical Mechanics:
  Newton’s Laws, Elasticity, Simple Harmonic Motion, Damped, Forced and Coupled Oscillations, Two- Body Dynamics, Centre of Mass, Reduced Mass, Collisions, Rotational Motion, Torque, Angular Momentum, Moment of Inertia, Central Forces, Gravitation, Kepler’s Laws
  
  Special Relativity:
  Lorentz Transformations, Length Contraction and Time Dilation, Paradoxes, Velocity Transformations, Relativistic Energy and Momentum
  
  Waves:
  Wave Equation, Travelling Waves, Superposition, Interference, Beats, Standing Waves, Dispersive Waves, Group Velocity, Doppler Effect
  
  Electricity and Magnetism:
  Static electric and magnetic fields. Time varying magnetic fields and motional emf. Electrical circuit analysis including dc and ac theory and circuit transients
  
  Light and Optics:
  Electromagnetic waves, dispersion by prisms and diffraction gratings, interference, diffraction, polarization, X-rays.
  
  Quantum Theory:
  Wave-particle duality, photoelectric effect, Bohr model, spectra of simple atoms, radioactive decay, fission and fusion, fundamental forces and the Standard Model.
  
  Thermodynamics:
  Kinetic theory of gases, Van der Waal’s equation, first and second laws of thermodynamics, internal energy, heat capacity, entropy. Thermodynamic engines, Carnot cycle. Changes of state.
  
  Solid State:
  Solids, crystal structure, bonding and potentials, thermal expansion. Introduction to band structure of metals, insulators and semiconductors. Origin and behaviour of electric and magnetic dipoles.

  Learning Outcomes

  Demonstrate knowledge and conceptual understanding in the areas of classical mechanics, special relativity, waves and oscillations, electricity and magnetism, light and optics, quantum theory, thermodynamics, and solid state, by describing, discussing and illustrating key concepts and principles.
  
  Solve problems by identifying relevant principles and formulating them with basic mathematical relations.
  
  Perform quantitative estimates of physical parameters within an order of magnitude.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Communicating scientific concepts in a clear and concise manner both orally and in written form. Working independently and with a group of peers. Time management and the ability to meet deadlines.

  Coursework

  0%

  Examination

  70%

  Practical

  30%

  Stage/Level

  1

  Credits

  40

  Module Code

  PHY1001

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  No

  Core/Optional

  Core

• Year 2

  Core Modules

  Linear Algebra (20 credits)

  ×

  Linear Algebra

  Overview

  - Recap and extend to fields such as C, the notions of abstract vector spaces and subspaces, linear independence, basis, dimension.
  - Linear transformations, image, kernel and dimension formula.
  - Matrix representation of linear maps, eigenvalues and eigenvectors of matrices.
  - Matrix inversion, definition and computation of determinants, relation to area/volume.
  - Change of basis, diagonalization, similarity transformations.
  - Inner product spaces, orthogonality, Cauchy-Schwarz inequality.
  - Special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties.
  - Basic computer application of linear algebra techniques.
  
  Additional topics and applications, such as: Schur decomposition, orthogonal direct sums and geometry of orthogonal complements, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module: have a good understanding and ability to use the basics of linear algebra; be able to perform computations pertaining to problems in these areas; have reached a good level of skill in manipulating basic and complex questions within this framework, and be able to reproduce, evaluate and extend logical arguments; be able to select suitable tools to solve a problem, and to communicate the mathematical reasoning accurately and confidently.

  Skills

  Analytic argument skills, computation, manipulation, problem solving, understanding of logical arguments.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2011

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Quantum & Statistical Physics (20 credits)

  ×

  Quantum & Statistical Physics

  Overview

  Quantum history, particle waves, uncertainty principle, quantum wells, Schrödinger wave equation SWE.
  
  1D SWE Solutions:
  Infinite and finite square potential well, harmonic potential well, particle wave at a potential step, particle wave at a potential barrier, quantum tunnelling, 1st order perturbation theory.
  
  3D Solutions of SWE:
  Particle in a box, hydrogen atom, degeneracy.
  
  Statistical Mechanics:
  Pauli exclusion principle, fermions, bosons, statistical distributions, statistical entropy, partition function, density of states. Examples of Boltzmann, Fermi-Dirac, Bose-Einstein distributions.

  Learning Outcomes

  Demonstrate how fundamental principles in quantum and statistical mechanics are derived and physically interpreted. In particular the uncertainty principle, the Schrödinger wave equation, tunnelling, quantum numbers, degeneracy, Pauli exclusion principle, statistical entropy, Boltzmann, Fermi-Dirac and Bose-Einstein distributions.
  
  Obtain and interpret solutions of the Schrödinger wave equation in 1D for several simple quantum wells and barriers, and in 3D for a particle in a box and the hydrogen atom.
  
  Apply quantum mechanics and statistical distributions to explain different physical phenomena and practical applications.
  
  Plan, execute and report the results of an experiment or investigation, and compare results critically with predictions from theory

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  40%

  Examination

  60%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  PHY2001

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Core

  Electricity, Magnetism and Optics (20 credits)

  ×

  Electricity, Magnetism and Optics

  Overview

  Electrostatics and magnetostatics.
  Coulomb, Gauss, Faraday, Ampère, Lenz and Lorentz laws
  Wave solution of the Maxwell’s equations in vacuum and the Poynting vector.
  Polarisation of E.M. waves and behaviour at plane interfaces.
  Propagation of light in media (isotropic dielectrics). Faraday and Kerr effects.
  Temporal and spatial coherence of light. Interference and diffraction
  Geometrical optics and matrix description of optic elements
  Optical cavities and laser action.

  Learning Outcomes

  Students will be able to:
  
  Define and describe the fundamental laws of electricity and magnetism, understand their physical significance, and apply them to well-defined physical problems.
  
  Formulate and manipulate Maxwell’s equations to obtain electromagnetic wave equations, solving them for propagation in vacuum, isotropic media, and at interfaces.
  
  Explain and formulate examples of optical phenomena such as interference, diffraction, Faraday and Kerr effects, laser action, and manipulation of light by optical components.
  
  Plan, execute and report the results of an experiment or investigation, and compare results critically with predictions from theory

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  40%

  Examination

  60%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  PHY2004

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Core

  Mathematical Methods 2 (20 credits)

  ×

  Mathematical Methods 2

  Overview

  Functions of a complex variable: limit in the complex plane, continuity, complex differentiability, analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, residues, Cauchy residue theorem, evaluation of integrals using the residue theorem.
  Series solutions to differential equations: Frobenius method.
  Fourier series and Fourier transform. Basis set expansion.
  Introduction to partial differential equations. Separation of variables. Wave equation, diffusion equation and Laplace’s equation.

  Learning Outcomes

  On completion of the module, the students are expected to be able to:
  • determine whether or not a given complex function is analytic;
  • recognise and apply key theorems in complex integration;
  • use contour integration to evaluate real integrals;
  • apply Fourier series and transforms to model examples;
  • solve the wave equation, diffusion equation and Laplace’s equation with model boundary conditions, and interpret the solutions in physical terms.

  Skills

  • Proficiency in complex calculus and its application to a range of problems.
  • Constructing and presenting mathematical and logical arguments.
  • Mathematical modelling and problem solving.
  • Ability to manipulate precise and intricate ideas.
  • Analytical thinking and logical reasoning.

  Coursework

  40%

  Examination

  60%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2021

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Optional Modules

  Group Theory (20 credits)

  ×

  Group Theory

  Overview

  - definition and examples of groups and their properties
  - countability of a group and index
  - Lagrange’s theorem
  - normal subgroups and quotient groups
  - group homomorphisms and isomorphism theorems
  - structure of finite abelian groups
  - Cayley’s theorem
  - Sylow’s theorem
  - composition series and solvable groups

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to: understand the ideas of binary operation, associativity, commutativity, identity and inverse; reproduce the axioms for a group and basic results derived from these; understand the groups arising from various operations including modular addition or multiplication of integers, matrix multiplication, function composition and symmetries of geometric objects; understand the concept of isomorphic groups and establish isomorphism, or otherwise, of specific groups; understand the concepts of conjugacy and commutators; understand the subgroup criteria and determine whether they are satisfied in specific cases; understand the concepts of cosets and index; prove Lagrange's theorem and related results; understand the concepts and basic properties of normal subgroups, internal products, direct and semi-direct products, and factor groups; establish and apply the fundamental results about homomorphisms - including the first, second and third isomorphism theorems - and test specific functions for the homomorphism property; perform various computations on permutations, including decomposition into disjoint cycles and evaluation of order; apply Sylow's theorem.

  Skills

  Numeracy and analytic argument skills, problem solving, analysis and construction of proofs.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2014

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Classical Mechanics (20 credits)

  ×

  Classical Mechanics

  Overview

  Introduction to calculus of variations.
  Recap of Newtonian mechanics.
  Generalised coordinates. Lagrangian. Least action principle. Conservation laws (energy, momentum, angular momentum), symmetries and Noether’s theorem. Examples of integrable systems. D’Alembert’s principle. Motion in a central field. Scattering. Small oscillations and normal modes. Rigid body motion.
  Legendre transformation. Canonical momentum. Hamiltonian. Hamilton’s equations. Liouville’s theorem. Canonical transformations. Poisson brackets.

  Learning Outcomes

  On completion of the module, the students are expected to be able to:
  • Derive the Lagrangian and Hamiltonian formalisms;
  • Demonstrate the link between symmetries of space and time and conservation laws;
  • Construct Lagrangians and Hamiltonians for specific systems, and derive and solve the corresponding equations of motion;
  • Analyse the motion of specific systems;
  • Identify symmetries in a given system and find the corresponding constants of the motion;
  • Apply canonical transformations and manipulate Poisson brackets.

  Skills

  • Proficiency in classical mechanics, including its modelling and problem-solving aspects.
  • Assimilating abstract ideas.
  • Using abstract ideas to formulate and solve specific problems.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2031

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Metric Spaces (20 credits)

  ×

  Metric Spaces

  Overview

  - definition and examples of metric spaces (including function spaces)
  - open sets, closed sets, closure points, sequential convergence, density, separability
  - continuous mappings between metric spaces
  - completeness

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to: understand the concept of a metric space; understand convergence of sequences in metric spaces; understand continuous mappings between metric spaces; understand the concepts and simple properties of special subsets of metric spaces (such as open, closed and compact); understand the concept of Hilbert spaces, along with the basic geometry of Hilbert spaces, orthogonal decomposition and orthonormal basis.

  Skills

  Analytic argument skills, problem solving, analysis and construction of proofs.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2013

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Analysis (20 credits)

  ×

  Analysis

  Overview

  Cauchy sequences, especially their characterisation of convergence. Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omitted). Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test. Mean value theorems including that of Cauchy, proof of l'Hôpital's rule, Taylor's theorem with remainder. Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to: understand and apply the Cauchy property together with standard Level 1 techniques and examples in relation to limiting behaviour for a variety of sequences; understand the relationships between sequences and series, especially those involving the Cauchy property, and of standard properties concerning absolute and conditional convergence, including power series and Taylor series; demonstrate understanding of the concept of uniform continuity of a real function on an interval, its determination by a range of techniques, and its consequences; understand through the idea of differentiability how to develop and apply the basic mean value theorems; describe the process of Riemann integration and the reasoning underlying its basic theorems including the fundamental theorem of calculus, and relate the concept to monotonicity and continuity.

  Skills

  Knowledge of core concepts and techniques within the material of the module. A good degree of manipulative skill, especially in the use of mathematical language and notation. Problem solving in clearly defined questions, including the exercise of judgment in selecting tools and techniques. Analytic and logical approach to problems. Clarity and precision in developing logical arguments. Clarity and precision in communicating both arguments and conclusions. Use of resources, including time management and IT where appropriate.

  Coursework

  10%

  Examination

  90%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2012

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Employability for Physics (0 credits)

  ×

  Employability for Physics

  Overview

  Introduction to placement for Physics students, CV building, international options, interview skills, assessment centres, placement approval, health & safety and wellbeing. Workshops on CV building and interview skills. This module is delivered in-house with the support of the QUB Careers Service and external experts.

  Learning Outcomes

  To identify gaps in personal employability skills. To plan a programme of work to result in a successful work placement application.

  Skills

  Plan self-learning and improve performance, as the foundation for lifelong learning/CPD. Decide on action plans and implement them effectively. Clearly identify criteria for success and evaluate their own performance against them .

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  2

  Credits

  0

  Module Code

  PHY2010

  Teaching Period

  Autumn

  Duration

  10 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Physics of the Solid State (20 credits)

  ×

  Physics of the Solid State

  Overview

  Periodicity and symmetry, basic crystallographic definitions, packing of atomic planes, crystal structures, the reciprocal lattice, diffraction from crystals, Bragg condition and Ewald sphere. Lattice waves and dispersion relations, phonons, Brillouin zones, heat capacity, density of vibrational states, Einstein and Debye models of heat capacity, thermal conductivity and anharmonicity. Concepts related to phase transitions in materials such as: free energy, enthalpy, entropy, order parameter, classification of phase transitions, Landau theory. Bonding classification of materials, metals, insulators and semiconductors, Fermi energy and density of electron states, energy bands, intrinsic and extrinsic semiconductors, donors and acceptors, carrier transport properties, p-n junction.

  Learning Outcomes

  Students will be able to:
  
  Recognise and define the fundamental concepts used to describe properties of the solid state such as simple crystal structures and symmetries, diffraction and the reciprocal lattice, vibrational and thermal properties, phase changes, and electrical properties, and to demonstrate conceptual understanding of these concepts.
  
  Show how relevant theoretical models can be developed to establish properties of materials and explain how these have been exploited in technological devices.
  
  Plan, execute and report the results of an experiment or investigation, and compare results critically with predictions from theory

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  40%

  Examination

  60%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  PHY2002

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  Astrophysics I (20 credits)

  ×

  Astrophysics I

  Overview

  Introduction to Astronomy: Units of measurement, telescopes and detecting photons.
  
  From planets to galaxies: Size and scale of the visible Universe, Stellar and galactic motion.
  
  The Solar system: The Sun as a star, Newtonian gravity; basic concepts in orbital dynamics, our solar system.
  
  Stars – observational properties/characterization: Stellar luminosities, colours, the Hertzsprung-Russell diagram, stellar classification, fundamental stellar properties, Stefan Boltzmann equation, mass-luminosity relations.
  
  Stars – stellar structure: Equation of hydrostatic support (including use of mass coordinate), gravitational binding and thermal energy of stars, Virial theorem, energy generation, energy transport by photon diffusion, convection.
  
  Stars – formation, stellar evolution, binary-star evolution, stellar death: single star evolution, post-H burning, binary-star evolution concepts and accretion, stellar end-states and compact objects.

  Learning Outcomes

  Students will be able to:
  
  Calculate photon fluxes and magnitudes for a sample of astrophysical sources.
  
  Understand the relative sizes of astrophysical objects and the standard units used to report them.
  
  Describe how the Hertzsprung-Russell diagram is constructed and physically interpreted.
  
  Use knowledge of physical concepts to derive simple equations that govern the internal structure of stars, and understand energy generation and transport in main-sequence stars, and how Kepler’s Laws originate from the gravitational forces.
  
  Comprehend how the observed properties of stars together with physical laws allow us to understand the evolution of stars of various masses.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  20%

  Examination

  40%

  Practical

  40%

  Stage/Level

  2

  Credits

  20

  Module Code

  PHY2003

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  Atomic and Nuclear Physics (20 credits)

  ×

  Atomic and Nuclear Physics

  Overview

  Atomic:
  Hydrogenic quantum numbers, Stern-Gerlach experiment, spin-orbit interaction, fine structure, quantum defect theory, central field approximation, LS coupling, Hund's rules, theory of the helium atom, selection rules, atomic spectra and transition probabilities, first order perturbation theory, Zeeman effect.
  
  Nuclear:
  Observation of nuclear properties, nuclear radius, mass (semi-empirical formula), inter-nucleon potential, radioactive decay mechanisms, fission and fusion, interactions of particles with matter.

  Learning Outcomes

  Students will be able to:
  
  Describe how atomic models have been developed from theoretical concepts and experimental observations.
  
  Recognise and use basic definitions to define atomic states and perform routine calculations to predict their energies and properties.
  
  Describe qualitatively the properties of nuclei and radiation making quantitative estimates of properties such as nuclear radius, binding energy, particle energy, and Q-values.
  
  Plan, execute and report the results of an experiment or investigation, and compare results critically with predictions from theory

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  40%

  Examination

  60%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  PHY2005

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  Employability for Mathematics (0 credits)

  ×

  Employability for Mathematics

  Overview

  Introduction to placement for mathematics and physics students, CV building, international options, interview skills, assessment centres, placement approval, health and safety and wellbeing. Workshops on CV building and interview skills. The module is delivered in-house with the support of the QUB Careers Service and external experts.

  Learning Outcomes

  Identify gaps in personal employability skills. Plan a programme of work to result in a successful work placement application.

  Skills

  Plan self-learning and improve performance, as the foundation for lifelong learning/CPD. Decide on action plans and implement them effectively. Clearly identify criteria for success and evaluate their own performance against them .

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  2

  Credits

  0

  Module Code

  MTH2010

  Teaching Period

  Autumn

  Duration

  10 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

• Year 3

  Core Modules

  Professional Skills (20 credits)

  ×

  Professional Skills

  Overview

  Development of oral presentation skills. Presentations to large groups/peers in a research or popular science context. Probing scientific understanding, critiquing presentations, peer review. Entrepreneurship, career guidance, CV writing, interview techniques. Essay writing and scientific writing skills

  Learning Outcomes

  Students will be able to:
  
  Search for, evaluate and reference relevant information from a range of sources
  
  Communicate general scientific topics in a clear and concise manner both orally and in a written format with proper regard for the needs of the audience.
  
  Critically question and evaluate the work of peers
  
  Critically self-reflect on progression of skills, academic performance, entrepreneurship and future prospects

  Skills

  Problem solving. Scientific writing. Entrepreneurship. Working independently and with a group of peers. Time management and the ability to meet deadlines.

  Coursework

  30%

  Examination

  0%

  Practical

  70%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3008

  Teaching Period

  Both

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Computational Projects (20 credits)

  ×

  Computational Projects

  Overview

  Computing coding skills and optimization techniques.
  Solution of ordinary differential equations with, for example, Runge Kutta 4th order method.
  Students to choose from a range of computational projects including projects to solve ordinary differential equations, for example in solution of the 1D time independent Schrödinger Equation with the Shooting method, and partial differential equations, for example simulation of a wave on a string.
  Data analysis techniques, for example, coping with noise and experimental uncertainty.

  Learning Outcomes

  Students will be able to:
  
  Analyse physical systems and write computer programs to model them.
  
  Use computational methods for robust analysis of experimental data.

  Skills

  Problem solving with computing methods and computer programming. Searching for and evaluating information from a range of sources. Communicating scientific concepts in a clear and concise manner both orally and in written form. Working independently and with a group of peers. Time management and the ability to meet deadlines.

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3009

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Investigations (20 credits)

  ×

  Investigations

  Overview

  Students conduct a short practice investigation, followed by two short investigations (in small groups and solo) in a range of problems in Applied Mathematics and Theoretical Physics. This is followed by a long investigation, which is a literature study of a Mathematical or Theoretical Physics topic not covered in the offered (or chosen) modules. The two short and the long investigation are typed up in reports and submitted for assessment.

  Learning Outcomes

  On completion of the module, it is intended that students will be able to:
  
  consider a problem or phenomenon and develop a mathematical model that describes it, stating any assumptions made;
  
  solve the model or its simplified version and analyse the results;
  
  suggest generalisations or extensions of the model to related problems or phenomena, and indicate possible ways of solving them;
  
  communicate the results of an investigation in a written (typed) report, with mathematical equations, tables, etc. as required, and illustrated by diagrams;
  
  investigate an unfamiliar topic using one or a number of literature sources, and write (type) a report that explains the topic in a logical manner, puts the topic in a wider context, uses equations, mathematical derivations, graphs and tables as necessary, and contains a bibliography list.

  Skills

  Research skills, presentational skills. Use of many sources of information.

  Coursework

  80%

  Examination

  0%

  Practical

  20%

  Stage/Level

  3

  Credits

  20

  Module Code

  AMA3020

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Numerical Analysis (20 credits)

  ×

  Numerical Analysis

  Overview

  • Introduction and basic properties of errors: Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.
  • Solution of equations in one variable: Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's "delta-squared" process; Roots of polynomials.
  • Solution of linear equations: LU decomposition; Pivoting strategies; Calculating the inverse; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.
  • Interpolation and polynomial approximation: Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.
  • Approximation theory: Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.
  • Numerical quadrature: Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature; Gaussian quadrature (Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite, Gauss-Chebyshev).
  • Numerical solution of ordinary differential equations: Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.

  Learning Outcomes

  On completion of the module, it is intended that students should: appreciate the importance of numerical methods in mathematical modelling; be familiar with, and understand the mathematical basis of, the numerical methods employed in the solution of a wide variety of problems;
  through the computing practicals and project, have gained experience of scientific computing and of report-writing using a mathematically-enabled word-processor.

  Skills

  Problem solving skills; computational skills; presentation skills.

  Coursework

  50%

  Examination

  50%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3023

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Quantum Theory (20 credits)

  ×

  Quantum Theory

  Overview

  • Overview of classical physics and the need for new theory.
  • Basic principles: states and the superposition principle, amplitude and probability, linear operators, observables, commutators, uncertainty principle, time evolution (Schrödinger equation), wavefunctions and coordinate representation.
  • Elementary applications: harmonic oscillator, angular momentum, spin.
  • Motion in one dimension: free particle, square well, square barrier.
  • Approximate methods: semiclassical approximation (Bohr-Sommerfeld quantisation), variational method, time-independent perturbation theory, perturbation theory for degenerate states (example: spin-spin interaction, singlet and triplet states).
  • Motion in three dimensions: Schrödinger equation, orbital angular momentum, spherical harmonics, motion in a central field, hydrogen atom.
  • Atoms: hydrogen-like systems, Pauli principle, structure of many-electron atoms and the Periodic Table.

  Learning Outcomes

  On the completion of this module, successful students will be able to
  • Understand, manipulate and apply the basic principles of Quantum Theory involving states, superpositions, operators and commutators;
  • Apply a variety of mathematical methods to solve a range of basic problems in Quantum Theory, including the finding of eigenstates, eigenvalues and wavefunctions;
  • Use approximate methods to solve problems in Quantum Theory and identify the range of applicability of these methods;
  • Understand the structure and classification of states of the hydrogen atom and explain the basic principles behind the structure of atoms and Periodic Table.

  Skills

  • Proficiency in quantum mechanics, including its modelling and problem-solving aspects.
  • Assimilating abstract ideas.
  • Using abstract ideas to formulate specific problems.
  • Applying a range of mathematical methods to solving specific problems.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3032

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Modelling and Simulation (20 credits)

  ×

  Modelling and Simulation

  Overview

  In this module, students will analyse real-life situations, build a mathematical model, solve it using analytical and/or numerical techniques, and analyse and interpret the results and the validity of the model by comparing to actual data. The emphasis will be on the construction and analysis of the model rather than on solution methods. Two group projects will fix the key ideas and incorporate the methodology. This will take 7-8 weeks of term and will be supported with seminars and workshops on the modelling process. Then students will focus on a solo project (relevant to their pathways) with real-life application and work individually on this for the remaining weeks of term. They will present their results in seminars with open discussion, and on a Webpage.
  The starting group project will be focused, and offer a limited number of specific modelling problems. For the other projects, students will build on these initial problems by addressing a wider problem taken from, but not exclusively, the following areas: classical mechanics, biological models, finance, quantum mechanics, traffic flow, fluid dynamics, and agent-based models, including modelling linked to problems of relevance to the UN sustainable development goals. A pool of options will be offered, but students will also have the opportunity to propose a problem of their own choice.

  Learning Outcomes

  On successful completion of the module, it is intended that students will be able to:
  
  1. Develop mathematical models of different kinds of systems using multiple kinds of appropriate abstractions
  2. Explain basic relevant numerical approaches
  3. Implement their models in Python and use analytical tools when appropriate
  4. Apply their models to make predictions, interpret behaviour, and make decisions
  5. Validate the predictions of their models against real data.

  Skills

  1. Creative mathematical thinking
  2. Formulation of models, the modelling process and interpretation of results
  3. Teamwork
  4. Problem-solving
  5. Effective verbal and written communication skills

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3024

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Quantum Mechanics and Relativity (20 credits)

  ×

  Quantum Mechanics and Relativity

  Overview

  Relativity:
  Einstein's postulates. The Lorentz transformation and consequences. 4-vector formulation. Relativistic particle dynamics. Relativistic wave dynamics. Relativistic electrodynamics.
  
  Quantum Mechanics:
  The Lagrangian and Hamiltonian formalism. Wavefunctions and operators. The Schrödinger equation. The harmonic oscillator. Three-dimensional systems: angular momentum. Three-dimensional system: spherical harmonics. Composition of angular momenta and spin. The Hydrogen atom. Special distributions: Bose-Einstein and Fermi-Dirac statistics. Bell inequality and quantum entanglement. Perturbation theory: time-independent perturbations. Perturbation theory: periodic perturbations

  Learning Outcomes

  Students will be able to:
  
  State the fundamental postulates of relativity and quantum mechanics, develop the mathematical formalism of these subjects.
  
  Solve specific physical problems using the formalism of relativity and quantum mechanics.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3001

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Optional Modules

  Dynamical Systems (20 credits)

  ×

  Dynamical Systems

  Overview

  Continuous dynamical systems
  - Fundamental theory: existence, uniqueness and parameter dependence of solutions;
  - Linear systems: constant coefficient systems and the matrix exponential; nonautonomous linear systems; periodic linear systems.
  - Topological dynamics: invariant sets; limit sets; Lyapunov stability.
  - Grobman-Hartman theorem.
  - Stable, unstable and centre manifolds.
  - Periodic orbits: Poincare-Bendixson theorem.
  - Bifurcations
  - Applications: the Van der Pol oscillator; the SIR compartmental model; the Lorenz system.
  Discrete dynamical systems
  - One-dimensional dynamics: the discrete logistic model; chaos; the Cantor middle-third set.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module: have a good understanding and ability to use the basics of dynamical systems; be able to perform computations pertaining to problems in these areas; have reached a good level of skill in manipulating basic and complex questions within this framework, and be able to reproduce, evaluate and extend logical arguments; be able to select suitable tools to solve a problem, and to communicate the mathematical reasoning accurately and confidently.

  Skills

  Analytic argument skills, computation, manipulation, problem solving, understanding of logical arguments.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3021

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Quantum Fields (20 credits)

  ×

  Quantum Fields

  Overview

  1. Review of fundamental quantum theory.
  2. Relativistic Lagrangian Field Theory.
  3. Symmetries and Conservation laws; Noether’s theorem
  4. The Klein-Gordon field. Second quantisation.
  5. The Dirac equation and the Dirac field. Spin-Statistics theorem.
  6. Covariant electromagnetic field quantisation.
  7. Interacting fields; the S-matrix; Wick’s theorem.
  8. Fundamentals of Quantum Electrodynamics; Feynman diagrams; probabilities of basic processes.

  Learning Outcomes

  On successful completion of the module, it is intended that students will be able to:
  1. Write relativistic Lagrangians for scalar, spinor and vector fields.
  2. Understand the fundamental role of symmetries and conservation laws for quantum fields.
  3. Describe the main features of the Klein-Gordon, Dirac and electromagnetic fields.
  4. Develop the S-matrix formalism for interacting fields.
  5. Apply the S-matrix to basic processes in quantum electrodynamics.
  6. Associate Feynman diagrams with scattering amplitudes and calculate related probabilities.
  7. Present complex ideas orally.

  Skills

  • Mathematical modelling
  • Problem solving
  • Abstract thinking

  Coursework

  20%

  Examination

  60%

  Practical

  20%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH4331

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Discrete Mathematics (20 credits)

  ×

  Discrete Mathematics

  Overview

  1. Intro Enumerative Combinatorics: basic counting, pigeonhole principle, inclusion-exclusion, recurrence relations, generating functions.
  2. Intro Elementary Number Theory: Divisibility and primes, Euclidean algorithm, linear congruences, Chinese Remainder Theorem.
  3. Intro Graph Theory: basic notions, trees, connectivity, matchings, graph colouring, planarity, basic Ramsey theory.
  4. Intro Algorithmics: analysis of algorithms, sorting, greedy and divide-and-conquer algorithms, basic graph and NT algorithms.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, demonstrate knowledge and confidence in applying key ideas and concepts of discrete mathematics, such as generating functions, the inclusion-exclusion principle, factorisation of integers, the Chinese Remainder Theorem and basic Ramsey theory. Students should be able to use concepts of graph theory to give precise formulations of problems and solve basic graph colouring problems.
  In addition, students should understand basic algorithmic principles such as binary trees, greedy algorithms and divide-and-conquer; they should be able to analyse the run time of basic algorithms such as the classical sorting algorithms by applying the techniques discussed in the first half of the module.

  Skills

  Knowing and applying basic techniques of discrete mathematics; formally analysing the run time of classical algorithms.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3022

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Geometry of Optimisation (20 credits)

  ×

  Geometry of Optimisation

  Overview

  • Functionals on R^n, linear equations and inequalities; hyperplanes; half-spaces
  • Convex polytopes; faces
  • Specific examples: e.g., traveling salesman polytope, matching polytopes
  • Linear optimisation problems; geometric interpretation; graphical solutions
  • Simplex algorithm
  • LP duality
  • Further topics in optimisation, e.g., integer programming, ellipsoid method

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to:
  • demonstrate understanding of the foundational geometry of convex polytopes;
  • demonstrate understand of the geometric ideas behind linear optimisation;
  • solve simple optimisation problems graphically;
  • apply the simplex algorithm to concrete optimisation problems.

  Skills

  Knowing and applying basic techniques of polytope theory and optimisation.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH4323

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Fourier Analysis and Applications to PDEs (20 credits)

  ×

  Fourier Analysis and Applications to PDEs

  Overview

  Introduction:
  - Examples of important classical PDEs (e.g. heat equation, wave equation, Laplace’s equation)
  - method of separation of variables
  Fourier series:
  - pointwise and L^2 convergence
  - differentiation and integration of Fourier series; using Fourier series to solve PDEs
  Distributions:
  - basic concepts and examples (space of test functions and of distributions, distributional derivative, Dirac delta)
  - convergence of Fourier series in distributions
  - Schwartz space, tempered distributions, convolution
  Fourier transform:
  - Fourier transform in Schwartz space, L^1, L^2 and tempered distributions
  - convolution theorem
  - fundamental solutions (Green’s functions) of classical PDEs

  Learning Outcomes

  On completion of the module it is intended that students will be able to:
  - use separation of variables to solve simple PDEs
  - understand the concept of Fourier series and be able to justify their convergence in various senses
  - find solutions of basic PDEs using Fourier series (including a justification of convergence)
  - understand the concept of distributions and tempered distributions
  - perform basic operations with distributions
  - understand the concept of Fourier transform in various settings
  - solve classical PDEs using Fourier transform (finding and using fundamental solutions)

  Skills

  Analytic argument skills, problem solving, use of generalized methods.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH4321

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Financial Mathematics (20 credits)

  ×

  Financial Mathematics

  Overview

  Introduction to financial derivatives: forwards, futures, swaps and options; Future markets and prices; Option markets; Binomial methods and risk-free portfolio; Stochastic calculus and random walks; Ito's lemma; the Black-Scholes equation; Pricing models for European Options; Greeks; Credit Risk.

  Learning Outcomes

  On completion of the module, it is intended that students will be able to: explain and use the basic terminology of the financial markets; calculate the time value of portfolios that include assets (bonds, stocks, commodities) and financial derivatives (futures, forwards, options and swaps); apply arbitrage-free arguments to derivative pricing; use the binomial model for option pricing; model the price of an asset as a stochastic process; define a Wiener process and derive its basic properties; obtain the basic properties of differentiation for stochastic calculus; derive and solve the Black-Scholes equation; modify the Black-Scholes equation for various types of underlying assets; price derivatives using risk-neutral expectation arguments; calculate Greeks and explain credit risk.

  Skills

  Application of Mathematics to financial modelling. Apply a range of mathematical methods to solve problems in finance. Assimilating abstract ideas.

  Coursework

  20%

  Examination

  70%

  Practical

  10%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3025

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Rings and Modules (20 credits)

  ×

  Rings and Modules

  Overview

  Rings, subrings, prime and maximal ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, principal ideal domains, modules, submodules and quotient modules, module maps, isomorphism theorems, chain conditions (Noetherian and Artinian), direct sums and products of modules, simple and semisimple modules.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to: understand, apply and check the definitions of ring and module; subring/submodule and ideal against concrete examples; understand and apply the isomorphism theorems; understand and check the concepts of integral domain, principal ideal domain and simple ring; understand and be able to produce the proof of several statements regarding the structure of rings and modules; master the concept of Noetherian and Artinian Modules and rings.

  Skills

  Numeracy and analytic argument skills, problem solving, analysis and construction of proofs.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3012

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Astrophysics II (20 credits)

  ×

  Astrophysics II

  Overview

  Advanced stellar structure and evolution: physics of stellar interiors; concepts of single-star evolution; end points of stellar evolution
  
  Radiative transfer: radiative transfer in solar and stellar atmospheres; statistical and ionization equilibrium, plasma diagnostics and line broadening processes
  
  Galaxies: the Milky Way galaxy; galaxy properties; physics of the interstellar medium, theories of galaxy formation and evolution

  Learning Outcomes

  Students will be able to:
  
  Demonstrate a detailed comprehension of the main concepts underpinning modern astrophysics with emphasis on stellar interiors/atmospheres, stellar evolution and galaxy structure / evolution.
  
  Explain the physics of stars and stellar evolution, and be able to describe the physical state of stars at all stages of their lives, and critically compare their fates and the various classes of objects they leave behind.
  
  Understand and be able to link the physical conditions existing in a variety of astrophysics environments, including stellar interiors, stellar atmospheres and galaxies to observations (including spectroscopy) and the principles of radiative transfer.
  
  Describe the properties of galaxies, their constituents and their evolution.
  
  Apply their knowledge to unfamiliar astrophysical problems.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3003

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  Advanced Solid State Physics (20 credits)

  ×

  Advanced Solid State Physics

  Overview

  Electrons in metals, including Drude classical theory, Sommerfeld quantum free electron model, nearly free electron model, effective mass, tightly bound electron model, Brillouin zones and energy bands, quantum wells and 2D electron gases, quantum hall effect, introductions to spintronics and superconductivity.
  Magnetism, including: underlying origin of magnetism, the link between dipole moment and angular momentum, diamagnetism, paramagnetism (classical and quantum treatments), ferromagnetism and the Weiss molecular field, antiferromagnetism.
  Dielectrics, including: concepts of polarization, polarisability, Mossotti field, contributions to polarization, the Mossotti catastrophe, ferroelectricity, soft mode descriptions of ferroelectricity and antiferroelectricity, Landau-Ginzburg-Devonshire theory, displacive versus order-disorder ferroelectrics.

  Learning Outcomes

  Students will be able to:
  
  Explain how lattice periodicity, structure and both classical and quantum mechanics lead to general concepts and observed properties of metals, dielectrics and magnetic materials.
  
  Formulate specific theoretical models of the properties of metals, dielectrics and magnetic materials and use these to make quantitative predictions of material properties.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3002

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  Advanced Electromagnetism and Optics (20 credits)

  ×

  Advanced Electromagnetism and Optics

  Overview

  Maxwell's equations, propagation of EM waves in dielectrics, conductors, anisotropic media, optical fibres/waveguides, non-linear optics. Polarisation, reflection and transmission at boundaries, Fresnel's equations. Thin/thick optical lenses, matrix methods, aberrations and diffraction.

  Learning Outcomes

  Students will be able to:
  
  Demonstrate knowledge and conceptual understanding of Maxwell's equations and their application to the propagation of electromagnetic waves in various media and their manipulation using optical components.
  
  Solve problems using mathematical techniques such as matrix methods and vector calculus to model electric/magnetic fields, the propagation of light, and to obtain analytical or approximate solutions.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3004

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  Nuclear and Particle Physics (20 credits)

  ×

  Nuclear and Particle Physics

  Overview

  Nuclear reaction classifications, scattering kinetics, cross sections, quantum mechanical scattering, nuclear shell model and scattering experiments, partial waves. Beta decay and neutrino mass, Fermi theory of beta decay and parity violation. Elementary particles; symmetry principles, unitary symmetry and quark model, particle interactions.

  Learning Outcomes

  Students will be able to:
  
  Show how theoretical concepts can be used to develop models of the nuclear structure, nuclear reactions, particle scattering, and beta decay, and report on supporting experimental evidence.
  
  Describe the principles of and evidence for the Standard Model
  
  Apply theoretical models to make quantitative estimates and predictions in nuclear and particle physics.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3005

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Measure and Integration (20 credits)

  ×

  Measure and Integration

  Overview

  - sigma-algebras, measure spaces, measurable functions
  - Lebesgue integral, Fatou's lemma, monotone and dominated convergence theorems
  - Fubini’s Theorem, change of variables theorem
  - Integral inequalities and Lp spaces

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to: understand the concepts of an algebra and a sigma-algebra of sets, additive and sigma-additive functions on algebras of sets, measurability of a function with respect to a sigma-algebra of subsets of the domain, integrability, measure and Lp-convergence of sequences of measurable functions; demonstrate knowledge and confidence in applying the Caratheodory extension theorem, Fatou's lemma and the monotone convergence theorem, the Lebesgue dominated convergence theorem, the Riesz theorem, Fubini’s theorem, change of variable’s theorem and integral inequalities; proofs excepting those of the Caratheodory and Riesz theorems; understand similarities and differences between Riemann and Lebesgue integration of functions on an interval of the real line.

  Skills

  Analytic argument skills, problem solving, analysis and construction of proofs.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3011

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Physics in Medicine (20 credits)

  ×

  Physics in Medicine

  Overview

  Fundamental principles, and technical and clinical applications of: interaction of electromagnetic radiation and ionising radiation with the body, lasers for therapy and imaging, ultrasound, radiation imaging techniques, radiotherapy, magnetic resonance imaging.

  Learning Outcomes

  Students will be able to:
  
  Describe, apply and discuss the underlying physical principles of techniques used for medical imaging techniques and treatment of diseased tissue with light and radiation.
  
  Evaluate the relative merits of current and future imaging and therapeutic techniques.
  
  Make quantitative estimates of relevant physical parameters such as penetration depth and radiation dose.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Communicating scientific concepts in a clear and concise manner both orally and in written form. Working independently and with a group of peers. Time management and the ability to meet deadlines.

  Coursework

  50%

  Examination

  50%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  PHY3006

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Classical Fields (20 credits)

  ×

  Classical Fields

  Overview

  • Recapping of the least action principle in Classical Mechanics. Lagrangians for continuous systems (e.g., a string), and derivation of the wave equation from Lagrange’s equation.
  • 4-dimensional space time, interval, 4-vectors, tensors, Lorentz covariance.
  • Action and Lagrangian for a particle, energy and momentum.
  • 4-potential and the Lagrangian for a charged particle in an electromagnetic field, relativistic equation of motion and Lorentz’s force, electric and magnetic fields.
  • Lagrangian of the electromagnetic field, Maxwell’s equations in covariant form, charge density and current density, continuity equation, Maxwell’s equations in conventional (3+1) form, and in integral form.
  • Electrostatics (general ideas, Coulomb’s law, fields of various charge distributions, electric dipole moment).
  • Magnetostatics (general ideas, Biot-Savart-Laplace law, fields of systems of currents, magnetic dipole moment).
  • Electromagnetic waves, plane wave, polarisation, monochromatic wave.
  • Electromagnetic radiation: retarded potentials, dipole radiation (electric, magnetic), Larmor formula.

  Learning Outcomes

  On the completion of this module, successful students will be able to
  
  • Understand the basic notions of 4-space, 4-vectors and covariance;
  • Derive the equations of the motions for the particle and for the fields (Maxwell equations) and solve them in a number of simple settings;
  • Derive the equations of electrostatics and magnetostatics from Maxwell’s equations and find their solutions for basic systems;
  •Understand the origins and nature of electromagnetic waves and determine the radiation by charges.

  Skills

  • Proficiency in classical fields, including its modelling and problem-solving aspects.
  • Assimilating abstract ideas.
  • Using abstract ideas and mathematical methods to formulate and solve specific problems.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3031

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

• Year 4

  Core Modules

  Physics Research Project (60 credits)

  ×

  Physics Research Project

  Overview

  Students will undertake a single research project within a Research Centre in the School or at an appropriate external organisation. Safety, risk assessment, and ethics training. Searching and evaluating scientific literature. Students will work full-time to complete all laboratory/computational results by the end of the first semester.

  Learning Outcomes

  Students will be able to:
  
  Plan, execute and report the results of an experiment or investigation, and compare critically with previous experiments or theory.
  
  Exploit computer technology to analyse and present data
  
  Demonstrate knowledge and understanding in a selected research topic in Physics, the current trends in this field, and developments at the frontiers of this subject
  
  Generate research results or technical innovations which could be included in a scientific publication
  
  Appreciate the importance of health and safety and scientific ethics, and perform a project risk assessment

  Skills

  Searching for and evaluating information from a range of sources. Communicating scientific concepts in a clear and concise manner both orally and in written form. Working independently and within a research group. Time management and the ability to meet deadlines.

  Coursework

  85%

  Examination

  0%

  Practical

  15%

  Stage/Level

  4

  Credits

  60

  Module Code

  PHY4001

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Project (40 credits)

  ×

  Project

  Overview

  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.

  Learning Outcomes

  On completion of this two-semester module, it is intended that students will be able to: undertake a substantial research project in which they increasingly take ownership of the planning and development of the work; work independently, under supervision; survey and use existing literature as a basis for their work; develop mathematical theory of models relevant to the project and where appropriate use or develop computer programs to advance the work and draw conclusions; give a coherent written account of the work undertaken, of its significance and of the outcomes of the research, in a technical report which is accessible to a range of interested readers; make a substantial oral presentation of the work undertaken, the results obtained and the conclusions drawn, to an audience not all of whom will be experts in the field of study.

  Skills

  Independent working. Oral and written presentational skills.

  Coursework

  80%

  Examination

  0%

  Practical

  20%

  Stage/Level

  4

  Credits

  40

  Module Code

  AMA4005

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Optional Modules

  Advanced Quantum Theory (20 credits)

  ×

  Advanced Quantum Theory

  Overview

  1. Review of fundamental quantum theory (Postulates of quantum mechanics; Dirac notation; Schrödinger equation; spin-1/2 systems; stationary perturbation theory).
  2. Coupled angular momenta: spin-1/2 coupling; singlet and triplet subspaces for two coupled spin-1/2 particles; Coupling of general angular momenta;
  3. Spin-orbit coupling; fine and hyperfine structures of the hydrogen atom.
  4. Time-dependent 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: Bose-Einstein condensation and superfluidity.

  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 time-dependent 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.

  Coursework

  0%

  Examination

  80%

  Practical

  20%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4031

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Practical Methods for Partial Differential Equations (20 credits)

  ×

  Practical Methods for Partial Differential Equations

  Overview

  Basics: solving first order ordinary differential equations, partial derivatives, surface, volume and line integrals, the Gauss theorem, Stokes' Theorem.
  Partial differential equations (PDE) and their relation to physical problems: heat conduction, flow of a liquid, wave propagation, Brownian motion.
  First order PDE in two variables: the method of characteristics, the transversality condition, quasilinear equations and shock waves, conservation laws, the entropy condition, applications to traffic flows.
  Second order linear PDEs: classification and canonical forms.
  The wave equation: d`Alembert’s solution, the Cauchy problem, graphical methods.
  The method of separation of variables: the wave and the heat equations.
  Numerical methods: finite differences, stability, explicit and implicit schemes, the Crank-Nicolson scheme, a stable explicit scheme for the wave equation.
  Practical: the students are offered to solve a heat and a wave equation using the method of separation of variables and a finite difference scheme.
  The Sturm-Liouville problem: a theoretical justification for the method of separation of variables. Simple properties of the Sturmian eigenvalues and eigenfunctions.
  Elliptic equations: the Laplace and Poisson equations, maximum principles for harmonic functions, separation of variables for Laplace equation on a rectangle.
  Green's functions: their definition and possible applications, Green’s functions for the Poisson equation, the heat kernel.

  Learning Outcomes

  On completion of the module, it is intended that students will be able to:
  understand the origin of PDEs which occur in mathematical physics, solve linear and quasilinear first order PDEs using the method of characteristics, classify and convert to a canonical form second order linear PDEs, solve numerically and using different methods the wave and the heat equations, as well as second order linear PDEs of a more general type, solve a Sturm-Liouville problem associated with a linear PDE and use the eigenfunctions to expand and evaluate its solution, understand the type of boundary conditions required by an elliptic PDE and solve it using the method of separation of variables, construct the Green's function for simple PDEs and use them to evaluate the solution.

  Skills

  Upon completion the student will have theoretical and practical skills for solving problems described by partial differential equations

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4024

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Information Theory (20 credits)

  ×

  Information Theory

  Overview

  Introduction to information theory. Basic modular arithmetic and factoring. Finite-field arithmetic. Random variables and some concepts of probabilities. RSA cryptography and factorisation. 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. A brief introduction to low-density parity-check codes.

  Learning Outcomes

  On completion of the module, it is intended that students will be able to: explain the security of and put in use the RSA protocol; understand how to quantify information and mutual information; motivate the use of uniquely decipherable and instantaneous codes; use Huffman encoding scheme for optical coding; use source extension to improve coding efficiency; prove Shannon noiseless coding theorem; understand the relation between mutual information and channel capacity; calculate the capacity of some basic channels; use basic error correction techniques for reliable transmission over noisy channels.

  Skills

  Problem solving skills; report writing skills; computing skills

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4022

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Geometry of Optimisation (20 credits)

  ×

  Geometry of Optimisation

  Overview

  • Functionals on R^n, linear equations and inequalities; hyperplanes; half-spaces
  • Convex polytopes; faces
  • Specific examples: e.g., traveling salesman polytope, matching polytopes
  • Linear optimisation problems; geometric interpretation; graphical solutions
  • Simplex algorithm
  • LP duality
  • Further topics in optimisation, e.g., integer programming, ellipsoid method

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to:
  • demonstrate understanding of the foundational geometry of convex polytopes;
  • demonstrate understand of the geometric ideas behind linear optimisation;
  • solve simple optimisation problems graphically;
  • apply the simplex algorithm to concrete optimisation problems.

  Skills

  Knowing and applying basic techniques of polytope theory and optimisation.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4323

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Applied Algebra and Cryptography (20 credits)

  ×

  Applied Algebra and Cryptography

  Overview

  - (finite) fields and rings of polynomials over them.
  - the division algorithm and splitting of polynomials.
  - ideals and quotient rings, (principal) ideal domains, with examples from rings of polynomials.
  - polynomials in several indeterminates, Hilbert’s basis theorem.
  - applications of algebra to cryptography (such as affine Hill ciphers, RSA, lattice cryptography, Diophantine equations).
  - optional topics may include Euclidean rings, unique factorisation domains, greatest common divisor domains.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to:
  understand the concept of a ring of polynomials over a (finite field);
  apply the factorisation algorithm;
  understand ideals, quotient rings and the properties of quotient rings;
  understand how algebra can be applied to cryptography and be able to encrypt messages using methods from the module.

  Skills

  Analytic argument skills, problem solving, analysis and construction of proofs.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4021

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Quantum Fields (20 credits)

  ×

  Quantum Fields

  Overview

  1. Review of fundamental quantum theory.
  2. Relativistic Lagrangian Field Theory.
  3. Symmetries and Conservation laws; Noether’s theorem
  4. The Klein-Gordon field. Second quantisation.
  5. The Dirac equation and the Dirac field. Spin-Statistics theorem.
  6. Covariant electromagnetic field quantisation.
  7. Interacting fields; the S-matrix; Wick’s theorem.
  8. Fundamentals of Quantum Electrodynamics; Feynman diagrams; probabilities of basic processes.

  Learning Outcomes

  On successful completion of the module, it is intended that students will be able to:
  1. Write relativistic Lagrangians for scalar, spinor and vector fields.
  2. Understand the fundamental role of symmetries and conservation laws for quantum fields.
  3. Describe the main features of the Klein-Gordon, Dirac and electromagnetic fields.
  4. Develop the S-matrix formalism for interacting fields.
  5. Apply the S-matrix to basic processes in quantum electrodynamics.
  6. Associate Feynman diagrams with scattering amplitudes and calculate related probabilities.
  7. Present complex ideas orally.

  Skills

  • Mathematical modelling
  • Problem solving
  • Abstract thinking

  Coursework

  20%

  Examination

  60%

  Practical

  20%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4331

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Fourier Analysis and Applications to PDEs (20 credits)

  ×

  Fourier Analysis and Applications to PDEs

  Overview

  Introduction:
  - Examples of important classical PDEs (e.g. heat equation, wave equation, Laplace’s equation)
  - method of separation of variables
  Fourier series:
  - pointwise and L^2 convergence
  - differentiation and integration of Fourier series; using Fourier series to solve PDEs
  Distributions:
  - basic concepts and examples (space of test functions and of distributions, distributional derivative, Dirac delta)
  - convergence of Fourier series in distributions
  - Schwartz space, tempered distributions, convolution
  Fourier transform:
  - Fourier transform in Schwartz space, L^1, L^2 and tempered distributions
  - convolution theorem
  - fundamental solutions (Green’s functions) of classical PDEs

  Learning Outcomes

  On completion of the module it is intended that students will be able to:
  - use separation of variables to solve simple PDEs
  - understand the concept of Fourier series and be able to justify their convergence in various senses
  - find solutions of basic PDEs using Fourier series (including a justification of convergence)
  - understand the concept of distributions and tempered distributions
  - perform basic operations with distributions
  - understand the concept of Fourier transform in various settings
  - solve classical PDEs using Fourier transform (finding and using fundamental solutions)

  Skills

  Analytic argument skills, problem solving, use of generalized methods.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4321

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Mathematical Methods for Quantum Information Processing (20 credits)

  ×

  Mathematical Methods for Quantum Information Processing

  Overview

  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.
  4. Quantum Communication protocols: quantum cryptography; cloning; teleportation; dense coding.
  5. Quantum computing: review of classical circuits and logic gates; quantum circuits and algorithms; implementation of quantum circuits on small prototypes of quantum computers (IBM Quantum Experience); examples of physical Hamiltonians implementing quantum gates.
  6. Theory of entanglement: basic notions and pure-state entanglement manipulation; detection of entanglement; measures of entanglement; entanglement and non-locality, Bell's inequality; multipartite entanglement.

  Learning Outcomes

  On completion of the module, it is intended that students will be able to:
  1. Express linear operators in terms of the Dirac notation; derive both the matrix and outer-product representation of linear operators in Dirac notation; recognise Hermitian, normal, positive and unitary operators, and put in use their respective basic properties; construct Kronecker products and functions of operators.
  2. Comprehend and express the postulates of quantum mechanics in Dirac notation; define projective measurements and calculate their outcome probabilities and output states; give examples of destructive and non-destructive projective measurements; prove the uncertainty principle for arbitrary linear operators; define positive-operator-valued measurement and use their properties to discriminate between non-orthogonal states; prove the no-cloning theorem for generic pure states.
  3. Explain the necessity of using the mixed-state description of quantum systems; define the density operator associated with an ensemble of pure states; express the postulate of quantum mechanics with the density operator formalism; distinguish pure and mixed states; describe two-level system in the Bloch sphere; geometrically describe generic n-level systems; calculate the partial trace and the reduced density operator of a tensor-product system.
  4. Describe the dynamics of a non-isolated quantum system with the formalism of completely-positive and trace preserving (CPTP) dynamical maps; derive the operator-sum representation of a CPTP map; give examples of CPTP maps.
  5. Demonstrate the most relevant communication protocols for pure states using the Dirac notation for states and linear operators: super-dense coding, quantum teleportation and quantum key distribution.
  6. Describe the basic properties of classical circuit for classical computing in terms of elementary logic gates; define the main model of quantum computation in terms of quantum circuits and gates; comprehend and construct basic quantum algorithms: Grover, Deutsch-Josza and Shor algorithms; construct, implement, and test small quantum circuits on prototypes of quantum computers (IBM Quantum Experience).
  7. Define quantum entanglement for pure and mixed states; identify entangled states; manipulate pure entangled state via LOCC operations; calculate the amount of entanglement in simple quantum systems; define Bell inequalities and calculate their violation; define the entanglement in multiple composite systems.

  Skills

  • Mathematical modelling of quantum systems, including problem solving aspects in the context of quantum technologies.
  • Assimilating abstract ideas.
  • Using abstract ideas to formulate specific problems.
  • Applying a range of mathematical methods to solving specific problems.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4023

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Topology (20 credits)

  ×

  Topology

  Overview

  - Definition and examples (natural, geometric and pathological)
  - Continuity and homeomorphisms
  - Compact, Connected, Hausdorff
  - Subspaces and product spaces
  - Introduction to homotopy, calculations and applications

  Learning Outcomes

  It is intended that students shall, on successful completion of this module, be able to: use effectively the notions of topological space, continuous function and homeomorphism and give examples thereof; state and use the basic properties of the product and subspace topologies; apply effectively the properties of connectedness, compactness, and Hausdorffness; understand the relation between metric and topological spaces; understand how topological maps are related via homotopy and apply homotopical calculations to examples.

  Skills

  Analytic argument skills, problem solving, analysis and construction of proofs.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4011

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Ultrafast Science (10 credits)

  ×

  Ultrafast Science

  Overview

  Interaction of ultrashort laser pulses with atoms and molecules, multiphoton processes, tunnelling ionisation, high harmonic generation, attosecond pulse generation and characterisation, molecular alignment, free electron lasers, pump-probe techniques, ultrafast processes

  Learning Outcomes

  Students will be able to:
  
  Demonstrate knowledge and understanding of the properties ultrashort, intense laser pulses, how they interact with atoms and molecules, and the techniques employed in experimental measurements.
  
  Develop simple models for laser-atom/molecule interactions and use them to make qualitative and quantitative predictions.
  
  Design a hypothetical apparatus or experimental campaign to study an ultrafast process
  
  Review scientific literature and report on current research topics individually or as part of a group.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently or as part of a group and meeting deadlines.

  Coursework

  30%

  Examination

  0%

  Practical

  70%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4011

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Planetary Systems (10 credits)

  ×

  Planetary Systems

  Overview

  Overview of Solar system structure. Properties of asteroids, comets, Trans-Neptunian Objects. Solar System evolution. Planetary System formation including molecular clouds, Jean’s mass, disc formation, angular momentum considerations. Protoplanetary disks – observed and theoretical structure and lifetimes, planet formation. Finding exoplanets. Exoplanet properties. Planet migration. Planetary interiors. Exoplanet theory and observation. Habitability.

  Learning Outcomes

  Students will be able to:
  
  Understand the structure of planetary systems and protoplanetary disks, and describe how they are formed through the comparison of observations and theory.
  
  Understand different techniques for exoplanet discovery and calculate the values of planetary system parameters required for this.
  
  Use knowledge of physics to constrain the orbital evolution of planets and their interior structure.
  
  Describe the observed properties of planetary atmospheres by combining measurements with theory, and explain how these properties allow possible habitats for life to be evaluated.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4005

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Medical Radiation Simulation (10 credits)

  ×

  Medical Radiation Simulation

  Overview

  Introduction to a basic Linux scientific computational environment. Introduction to Monte-Carlo radiation transport simulation. Proton and photon interactions with matter. Applications of radiation transport to simulate aspects of medical imaging and radiotherapy. Validation of simulations and assessment of errors.

  Learning Outcomes

  Students will be able to:
  Solve a range of problems computationally involving the transport of radiation through matter, including assessing the validity of and errors associated with such simulations.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently or as part of a group and meeting deadlines.

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4004

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Ionising Radiation in Medicine (10 credits)

  ×

  Ionising Radiation in Medicine

  Overview

  Interactions of radiation with matter; Introduction to radio-biology; Interaction of Charged Particles with Biological Matter; Modern approaches to Radiotherapy; Selected Modern Radiation Research Topics

  Learning Outcomes

  Students will be able to:
  
  Comprehend the basis for radiation-based physical measurements pertinent to the human body
  
  Appreciate the role of modern radiation medical devices and the underlying physics at work
  
  Analyse and quantify the physical processes at work in a range of medical applications.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  70%

  Examination

  0%

  Practical

  30%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4003

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  High Energy Astrophysics (10 credits)

  ×

  High Energy Astrophysics

  Overview

  Observational overview
  Accreting neutron stars and pulsars
  Pulsar emission mechanisms
  Black holes, active galactic nuclei, explosive transients (gamma-ray bursts, supernovae), and supernova remnants
  Role of jets
  Non-electromagnetic processes; cosmic rays, gravitational waves
  Particle acceleration
  Radiation processes (e.g., Bremsstrahlung, inverse Compton, etc.)
  Stellar dynamos
  Flux emergence
  Magnetic topologies
  Zeeman + Hanle effects
  Magnetic reconnection and flares

  Learning Outcomes

  Students will be able to:
  
  Apply their knowledge of mathematics and physics from Levels 1-3 in an astrophysical context.
  
  Understand the evolutionary history of binary systems containing compact degenerate objects;
  
  Understand how high energy processes such as accretion and angular momentum transfer come into play in a variety of astrophysical objects on vastly different scales.
  
  Develop a sense of relevant observational signatures of high energy astrophysical processes that may be both electromagnetic and non-electromagnetic in nature.
  
  Critically compare the evidence from observations with the predictions from theory

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4006

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Laser Physics (10 credits)

  ×

  Laser Physics

  Overview

  Basic laser physics: Population inversion and laser materials, gain in a laser system, saturation, transform limit, diffraction limit
  
  Short pulse oscillators: Cavities, Q-switching, cavity modes, mode locking
  
  Amplification: Beam transport considerations (B-Integral), chirped pulse amplification, stretcher and compressor design, white light generation, optical parametric chirped pulse amplification.
  
  Different types of lasers: Fiber lasers, laser diodes, Dye lasers, high performance national and international laser facilities
  
  Applications of state of the art lasers: Intense laser-matter interactions, high harmonic generation : perturbed atoms to relativistic plasmas, generation of shortest pulses of electromagnetic radiation

  Learning Outcomes

  Students will be able to:
  
  Demonstrate knowledge and understanding of the basics of modern laser systems, and how the unique properties of the high power lasers and recent technological advances are opening up new research fields including next generation particle and light sources.
  
  Correlate the fundamental parameters of specific lasers or laser facilities to potential applications or research projects.
  
  Review published material on topics of high intensity laser-plasma interactions

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently or as part of a group and meeting deadlines.

  Coursework

  60%

  Examination

  40%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4007

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  The Physics of Nanomaterials (10 credits)

  ×

  The Physics of Nanomaterials

  Overview

  Physics of nanomaterials with the emphasis on fabrication of materials and applications in magnetic recording and photonics. Magnetic recording materials including bit patterned media and spin valves. Nanostructures for surface plasmon detection. Optical properties of metal nanoparticles and nanostructures. Concept of metamaterials and negative refractive index materials. Examples of applications of nanophotonic devices e.g. in imaging, sensing and data storage.

  Learning Outcomes

  Students will be able to:
  
  Demonstrate knowledge and understanding of physical principles underpinning nanostructured materials and of nano-optics and its applications.
  
  Identify design and propose fabrication routes to create nanostructured materials for various applications.
  
  Review and discuss scientific literature, and report on current research topics individually or as part of a group.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently or as part of a group and meeting deadlines.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4010

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Physics of Materials Characterisation (10 credits)

  ×

  Physics of Materials Characterisation

  Overview

  Fundamental physics underlying electron microscopy-based analysis to investigate the delicate link between crystal structure and chemical composition at the nanoscale, and its impact on properties, with special focus on functional oxides and semiconductors. Physical principles of spectroscopy, Infrared and Raman spectroscopy/microscopy, Scanning nonlinear optical microscopy and scanning probe microscopy with specific applications towards study of phase transitions, domains and ferroic materials.

  Learning Outcomes

  Students will be able to:
  
  Demonstrate knowledge and understanding of physical principles underpinning different spectroscopy and microscopy techniques relevant to study of phase transitions, ferroic materials and semiconductors.
  Identify, design and propose microscopy based experimental setups to study physical phenomena in solid state.
  Review and discuss scientific literature, and report on current research topics individually or as part of a group.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently or as part of a group and meeting deadlines.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4009

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Plasma Physics (10 credits)

  ×

  Plasma Physics

  Overview

  Introduction to Plasmas: applications, fundamental concepts
  Single particle orbit theory: Motion of charged particles in constant/varying electric and magnetic fields, particle drift
  Plasma as Fluid: Two fluids model, Plasma oscillations and frequency.
  Waves in Plasma: Electron plasma wave, Ion acoustic wave, electromagnetic wave propagation in plasma
  Collisions and Resistivity: Concept of plasma resistivity, Collisional absorption of laser in plasma
  Intense laser plasma Interaction: Resonance absorption, Landau damping, Ponderomotive force, Interaction in the relativistic regime, particle (electron and ion) acceleration mechanisms

  Learning Outcomes

  Students will be able to:
  Demonstrate knowledge and understanding of the physics of plasmas relevant to a range of research areas from astrophysics to laser-plasma interactions.
  Understanding and derive the behaviour of charges particles in presence of electric and magnetic fields.
  Derive and interpret various plasma phenomenon using fluid theory
  Review scientific literature and report on current research topics individually or as part of a group.

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently or as part of a group and meeting deadlines.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4008

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Cosmology (10 credits)

  ×

  Cosmology

  Overview

  Observational overview
  Distance scale and redshift
  Friedmann equation and expansion, and Universal geometry
  Cosmological models
  Observational parameters
  The cosmological constant
  Age of the universe
  Density of the universe and dark matter
  Cosmic microwave background
  Early universe
  Nucleosynthesis – the origin of light elements
  Inflationary universe and the Initial singularity

  Learning Outcomes

  Students will be able to:
  
  Apply their knowledge of basic physics including thermodynamics, atomic physics and nuclear physics to understand the principles of modern cosmology.
  
  Appreciate the concepts of the expanding Universe, redshift, isotropy and the mass energy content of the Universe.
  
  Formulate and manipulate equations of Newtonian gravity to derive the Friedmann equation. Solve this equation to obtain simple cosmological models.
  
  Explain the origin of the cosmic microwave background and the nucleosynthesis of the light elements in the big bang theory.
  
  Understand how precision measurements constrain the Hubble parameter, the age and the matter and energy density of the Universe. Understand the observational evidence for dark matter and the accelerating expansion.
  
  Critically compare the evidence from observations with the predictions from theory

  Skills

  Problem solving. Searching for and evaluating information from a range of sources. Written and oral communication of scientific concepts in a clear and concise manner. Working independently and meeting deadlines.

  Coursework

  50%

  Examination

  0%

  Practical

  50%

  Stage/Level

  4

  Credits

  10

  Module Code

  PHY4016

  Teaching Period

  Spring

  Duration

  6 weeks

  Pre-requisite

  No

  Core/Optional

  Optional