This degree combines the best aspects of the separate mathematics and physics degrees, offering enhanced flexibility, increased choice and the opportunity to pursue some of the most interesting and relevant questions that are pervasive in society, technology, the world and, indeed, the universe.
This degree provides a unique combination of factors: pure science with infinite intrigue awaiting a curious mind, and a hugely valuable and employable skill set enabling a broad range of possible future careers.
Physics at Queen's was ranked 3rd in the country for research intensity in the United Kingdom's most recent Research Excellence Framework (REF) exercise, as published by the Times Higher Education.
Applied Mathematics and Physics highlights
Global Opportunities
We participate in the IAESTE and Turing student exchange programmes, which enable students to obtain work experience in companies and universities throughout the world.
Industry Links
All students in the school have the option to include a year in industry as part of their studies. This is a fantastic opportunity to see mathematics at work in the real world, and to enhance your career prospects at the same time. Graduate employers include: BT; Seagate; Allstate; Randox; Andor; Civil Service.
World Class Facilities
The school has its own dedicated teaching centre which opened in September 2016. This building houses lecture and group-study rooms, a hugely popular student social area and state-of-the-art computer and laboratory facilities. The centre is an exciting hub for our students and is situated directly adjacent to the Lanyon Building on the main university campus. This makes us the only school with a dedicated teaching space right at the heart of the university.
Professional Accreditations
This degree is recognised by the Institute of Physics (IoP).
Internationally Renowned Experts
All of our faculty staff are research scientists in their own right; in the 2021 REF peer-review exercise, Physics Research Power was in the top 20 in the UK and Mathematics Research has the 11th highest impact in the UK.
Student Experience
In the 2020 National Student Survey, the Applied Maths and Physics degree had a 100% student satisfaction rating.
Further Study Opportunities
Placement Year:
Students can take an optional placement year between years 2/3 or years 3/4 of their course. Completion of an approved placement will be acknowledged in your final degree certificate with the addition of the words "with placement year".
Student Experience
School has the 3rd highest postgraduate research student satisfaction in the university.
Career Development
87% of Maths students are in graduate employment or further study 15 months after graduation (11th in the UK).
The most recent HESA data shows that over 95% of QUB physics graduates are in employment or further study 15 months after graduation.
Student Experience
In the 2023 National Student Survey physics scored above the benchmark in 6 out of 7 themes with a 94.9% positivity score on how well staff explained things.
The School of Mathematics and Physics was 3rd of 15 schools in the University in overall NSS score.
Student Testimonials
All of my lecturers seemed so enthusiastic about what they were teaching us. It was easy to feel the same way. And they were always very approachable. They always seemed really happy if you came to ask a question. In our first year, we also had weekly physics tutorials in small groups of three or four that were also really helpful. I didn’t expect such focused teaching.
My degree has given me a lot of wonderful opportunities. After I left QUB I went to Munich to do a PhD and now I’m in Paris working as a postdoctoral researcher. Everyone I graduated with seems to have done really well.
Dr. Aoife Boyle (MSci Physics with Astrophysics)
Studying physics at QUB has been a great way to gain a better understanding across a wide range of topics. Throughout the course we’ve also developed strong problem solving, computational and data analysis skills, which we can apply to wide range of jobs.”
Daniel Molloy (MSci Physics 2021)
The lecturers for Maths are fantastic. People who are incredibly enthusiastic about what they are teaching and about their own research fields.
Everyone I know who was in my year got solid (well-paid) jobs or post-graduate degrees within a year of graduating.
Michael Hart (MSci Mathematics 2019)
What set Maths apart from the rest was that I loved how open-ended it was. Instead of going into a degree that was niche, vocational and targeted at one area of study, I loved how many open doors a degree in Maths could offer me. It was a degree that is widely respected and recognised by many industries and organisations. I knew it would be a degree I could be proud of.
Overall, I am very pleased I decided to study Maths as a degree at QUB. I learnt so much throughout my time there, developed old and new skills, sparked an interest and passion within me for data and its analysis and give me a brilliant foundation to a career that has no limitations and endless opportunities.
The course unit details given below are subject to change, and are the latest example of the curriculum available on this course of study.
Stage 1
At Stage 1, students must take the four compulsory modules.
Stage 2
At Stage 2, students must take four compulsory modules plus two optional modules as approved by an advisor of studies.
Stage 3
At Stage 3, students must take modules totalling 120 units as approved by an advisor of studies. The choice must include at least 2 taught modules from Physics and 2 taught modules from Mathematics.
Stage 4
At Stage 4, students must take modules totalling 120 units as approved by an advisor of studies. The choice must include at least 2 taught modules from Physics and 2 taught modules from Mathematics plus a project module from either subject area.
School of Maths & Physics Dr Kar is a Reader in Physics and is an internationally recognised expert in the areas of high-intensity laser-plasma interaction. His main focus is on the development and optimisation of laser-driven ion and neutron sources for their wide-ranging applications in Science, security and healthcare.
School of Maths & Physics Dr Huettemann is a Senior Lecturer in Mathematics with research interests in homological algebra, graded algebra and K-theory.
Contact Teaching Hours
Personal Study
18 (hours maximum) 15-18 hours studying and revising in your own time each week, including some guided study using handouts, online activities, homeworks etc.
Large Group Teaching
9 (hours maximum) 9 hours of lectures.
Medium Group Teaching
5 (hours maximum) 5 hours of mathematics/physics and computer workshops each week in Level 1, with an average of 4 hours of practical work per week in Level 2 plus mathematical study classes.
Small Group Teaching/Personal Tutorial
2 (hours maximum) 2 hours of physics and mathematics tutorials/assignment classes (or later, project supervision).
Learning and Teaching
At Queen’s, we aim to deliver a high quality learning environment that embeds intellectual curiosity, innovation and best practice in learning, teaching and student support to enable students to achieve their full academic potential.
The MSci in Applied Mathematics and Physics provides a range of learning experiences which enable students to engage with subject experts, develop attributes and perspectives that will equip them for life and work in a global society and make use of innovative technologies and a world-class library that enhances their development as independent, lifelong learners. Examples of the opportunities provided for learning on this course are:
Computer based modules
These provide students with the opportunity to develop technical skills and apply theoretical principles to real-life or practical contexts.
E-learning technologies
Information associated with lectures and assignments is often communicated via a Virtual Learning Environment (VLE) called Canvas. A range of e-learning experiences are also embedded in the degree programme through the use of, for example, interactive support materials and web-based learning activities.
Laboratory physics
All students will undertake experimental physics as part of their degree. Students normally work in assigned pairs in the laboratory, with submitted reports and findings individually assessed. As part of this work students will become proficient in using Excel for analysing data and Word for laboratory reports.
Lectures
These introduce basic information about new topics as a starting point for further self-directed private study/reading. Lectures also provide opportunities to ask questions, gain some feedback and advice on assessments (normally delivered in large groups to all year group peers).
Personal tutor
Undergraduates are allocated a Personal Tutor during Level 1 and Level 2 who meets with them on several occasions during the year to support their academic development.
Self-directed study
This is an essential part of life as a Queen’s student when important private reading, engagement with e-learning resources, reflection on feedback to date and assignment research and preparation work is carried out.
Supervised projects
In final year, students will be expected to carry out a significant piece of research on a topic or practical methodology that they have chosen. Students will receive support from a supervisor who will guide them in terms of how to carry out research and who will provide feedback on at least 2 occasions during the write up stage.
Tutorials
Significant amounts of teaching are carried out in small groups (typically 10-20 students). These provide an opportunity for students to engage with academic staff who have specialist knowledge of the topic, to ask questions of them and to assess their own progress and understanding with the support of peers.
Assessment
The way in which students are assessed will vary according to the learning objectives of each module. Some modules are assessed solely through project work or written assignments. Others are assessed through a combination of coursework and end of semester examinations. Details of how each module is assessed are shown in the Student Handbook which may be accessed online via the School website.
Student Tutorial Questions/ Lecture Assignments
This involves the completion and submission of example problems on a weekly (tutorial) or three-weekly (assignment) basis as answered by individual students. These are submitted by students by an appropriate deadline and assessed, with the mark awarded contributing to the continuous assessment element of the module mark. The mark awarded reflects accuracy and clarity of the submitted answers together with understanding of the subject matter. Consistent with employer feedback, some modules also require students to prepare and make a small group presentations on a pre-assigned topic. Such group activities are also assessed, with the mark awarded contributing to the continuous assessment element of the module mark. To aid such exercises all students in their first year are given instruction and guidance on making successful presentations.
Laboratory and Computational Skills
All physics students are required to learn and understand the basic concepts of experimental physics. This involves understanding the basics of measurements, accuracy and error analysis; being able to understand and (in later levels) assess different methods of performing experimental measurements; reporting experimental findings and comparing them with prior knowledge of expectations based on physical laws. Assessment tales place through short laboratory reports or presentations, for which instruction is given. Additionally, all students will be given training in software coding using computer languages appropriate for scientific investigations, and this is assessed through worksheets and assignments.
Examinations
Most modules require the sitting of an unseen examination, to assess individual understanding of mathematical and physical concepts and the ability to tackle problems in the areas specific to the taught modules.
Feedback
As students progress through their course at Queen’s they will receive general and specific feedback about their work from a variety of sources including lecturers, module co-ordinators, placement supervisors, personal tutors, advisers of study and your peers. University students are expected to engage with reflective practice and to use this approach to improve the quality of their work. Feedback may be provided in a variety of forms including:
Feedback provided via formal written comments and marks relating to work that students, as individuals or as part of a group, have submitted.
Face to face comment. This may include occasions when students make use of the lecturers’ advertised “office hours” to help address a specific query.
Placement employer comments or references
Online or emailed comment
General comments or question and answer opportunities at the end of a lecture, seminar or tutorial
Pre-submission advice regarding the standards you should aim for and common pitfalls to avoid. In some instances, this may be provided in the form of model answers or exemplars which students can review in their own time
Feedback and outcomes from practical classes.
Comment and guidance provided by staff from specialist support services such as, Careers, Employability and Skills or the Learning Development Service
Once students have reviewed their feedback, they are encouraged to identify and implement further improvements to the quality of their work
Facilities
Undergraduate Teaching Centre
Throughout their time with us, students will use the new Mathematics and Physics Teaching Centre. Opened in 2016 with almost £2 million of new equipment, students can use the well-equipped twin computer rooms for both self-study and project work. This includes a small astronomical observatory on the roof of the main building. In the physics laboratories, students will be able to investigate everything from the nature of lasers, to the quantum mechanical properties of the electron, to the dynamic hydrogen chromosphere of the Sun.
The information below is intended as an example only, featuring module details for the current year of study (2024/25). Modules are reviewed on an annual basis and may be subject to future changes – revised details will be published through Programme Specifications ahead of each academic year.
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
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.
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.
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.
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.
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.
- 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.
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.
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.
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.
- 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.
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.
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.
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, thermal expansion and anharmonicity.
Concepts related to phase transitions in materials such as: free energy, enthalpy, entropy, order parameter, classification of phase transitions, Landau theory.
Electronic band structure, including: failures of classical model for metals and semiconductors, free electron gas description of metals, density of states, Fermi Dirac statistics, electronic heat capacity, development of band structure, prediction of intrinsic semiconducting behaviour, doping
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.
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.
- 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.
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 .
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 .
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.
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.
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.
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
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.
• 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.
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.
• 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.
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.
1. Simplicial complexes
2. PL functions
3. Simplicial homology
4. Filtrations and barcodes
5. Matrix reduction
6. The Mapper Algorithm
7. Learning with topological descriptors
8. Statistics with topological descriptors
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 topological data analysis, such as simplicial complexes, simplicial homology, barcodes, matrix reduction and the analysis of topological descriptors.
In addition, students should be able to use standard software (e.g. the freely available R package TDA) to analyse simple data sets.
Skills
Knowing and applying basic techniques of topological data analysis. In particular, this includes the analysis and interpretation of topological invariants of data sets; the production of graphical representations of such descriptors; and basic computational aspects of linear algebra.
A characterisation of finite-dimensional normed spaces; the Hahn-Banach theorem with consequences; the bidual and reflexive spaces; Baire’s theorem, the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-Steinhaus theorem; weak topologies and the Banach-Alaoglu theorem; spectral theory for bounded and compact linear operators.
Learning Outcomes
It is intended that students shall, on successful completion of the module, be able to: recognise when a normed space is finite dimensional; determine when linear functionals on normed spaces are bounded and determine their norms; be familiar with the basic theorems of functional analysis (Hahn-Banach, Baire, open mapping, closed graph and Banach-Steinhaus theorems) and be able to apply them; understand dual spaces, recognise the duals of the standard Banach spaces and recognise which of the standard Banach spaces are reflexive; understand the relations between weak topologies on normed spaces and compactness properties; be familiar with the basic spectral theory of bounded and compact linear operators.
Skills
Analysis of proof and development of mathematical techniques in linear infinite dimensional problems.
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.
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.
• 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.
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.
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.
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.
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.
Nuclear reaction classifications, scattering kinetics, cross sections, quantum mechanical scattering, Scattering experiments and the nuclear shell model, the inter-nucleon force, partial waves. Fermi theory of beta decay. Nuclear astrophysics and nuclear fission power generation. 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.
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.
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.
Electronic transport in metals, including: Lorentz-Drude classical theories and the Sommerfeld quantum free electron model. Influence of band structure on electron dynamics and transport. Electron scattering.
Magnetotransport, including cyclotron resonance, magnetoresistance and Hall effect
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.
- 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.
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.
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.
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.
1. Simplicial complexes
2. PL functions
3. Simplicial homology
4. Filtrations and barcodes
5. Matrix reduction
6. The Mapper Algorithm
7. Learning with topological descriptors
8. Statistics with topological descriptors
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 topological data analysis, such as simplicial complexes, simplicial homology, barcodes, matrix reduction and the analysis of topological descriptors.
In addition, students should be able to use standard software (e.g. the freely available R package TDA) to analyse simple data sets.
Skills
Knowing and applying basic techniques of topological data analysis. In particular, this includes the analysis and interpretation of topological invariants of data sets; the production of graphical representations of such descriptors; and basic computational aspects of linear algebra.
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.
- 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.
- (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.
A characterisation of finite-dimensional normed spaces; the Hahn-Banach theorem with consequences; the bidual and reflexive spaces; Baire’s theorem, the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-Steinhaus theorem; weak topologies and the Banach-Alaoglu theorem; spectral theory for bounded and compact linear operators.
Learning Outcomes
It is intended that students shall, on successful completion of the module, be able to: recognise when a normed space is finite dimensional; determine when linear functionals on normed spaces are bounded and determine their norms; be familiar with the basic theorems of functional analysis (Hahn-Banach, Baire, open mapping, closed graph and Banach-Steinhaus theorems) and be able to apply them; understand dual spaces, recognise the duals of the standard Banach spaces and recognise which of the standard Banach spaces are reflexive; understand the relations between weak topologies on normed spaces and compactness properties; be familiar with the basic spectral theory of bounded and compact linear operators.
Skills
Analysis of proof and development of mathematical techniques in linear infinite dimensional problems.
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
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.
• 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
AAA (Mathematics and Physics)
OR
A* (Mathematics) AB (Physics)
A maximum of one BTEC/OCR Single Award or AQA Extended Certificate will be accepted as part of an applicant's portfolio of qualifications with a Distinction* being equated to a grade A at A-level and a Distinction being equated to a grade B at A-level.
Irish leaving certificate requirements
H2H2H3H3H3H3 including Higher Level grade H2 in Mathematics and Physics
Access Course
Not considered. Students should apply for the BSc Applied Mathematics and Physics degree.
International Baccalaureate Diploma
36 points overall including 6 6 6 at Higher Level including Mathematics and Physics.
Graduate
A minimum of a 2:2 Honours Degree, provided any subject requirements are also met.
Note
All applicants must have GCSE English Language grade C/4 or an equivalent qualification acceptable to the University.
How we choose our students
Applications are dealt with centrally by the Admissions and Access Service rather than by the School of Mathematics and Physics. Once your on-line form has been processed by UCAS and forwarded to Queen's, an acknowledgement is normally sent within two weeks of its receipt at the University.
Selection is on the basis of the information provided on your UCAS form. Decisions are made on an ongoing basis and will be notified to you via UCAS.
For entry last year, applicants for programmes in the School of Mathematics and Physics offering A-level/BTEC Level 3 qualifications must have had, or been able to achieve, a minimum of five GCSE passes at grade C/4 or better (to include English Language and Mathematics), though this profile may change from year to year depending on the demand for places. The Selector also checks that any specific entry requirements in terms of GCSE and/or A-level subjects can be fulfilled.
Offers are normally made on the basis of three A-levels. The offer for repeat candidates may be one grade higher than for first time applicants. Grades may be held from the previous year.
Applicants offering two A-levels and one BTEC Subsidiary Diploma/National Extended Certificate (or equivalent qualification) will also be considered. Offers will be made in terms of the overall BTEC grade awarded. Please note that a maximum of one BTEC Subsidiary Diploma/National Extended Certificate (or equivalent) will be counted as part of an applicant’s portfolio of qualifications. The normal GCSE profile will be expected.
For applicants offering the Irish Leaving Certificate, please note that performance at Irish Junior Certificate (IJC) is taken into account. For last year’s entry, applicants for this degree must have had a minimum of five IJC grades at C/Merit. The Selector also checks that any specific entry requirements in terms of Leaving Certificate subjects can be satisfied.
Applicants offering other qualifications will also be considered. The same GCSE (or equivalent) profile is usually expected of those candidates offering other qualifications.
The information provided in the personal statement section and the academic reference together with predicted grades are noted but, in the case of degree courses in the School of Mathematics and Physics, these are not the final deciding factors in whether or not a conditional offer can be made. However, they may be reconsidered in a tie break situation in August.
A-level General Studies and A-level Critical Thinking would not normally be considered as part of a three A-level offer and, although they may be excluded where an applicant is taking four A-level subjects, the grade achieved could be taken into account if necessary in August/September.
Candidates are not normally asked to attend for interview.
If you are made an offer then you may be invited to a Faculty/School Visit Day, which is usually held in the second semester. This will allow you the opportunity to visit the University and to find out more about the degree programme of your choice and the facilities on offer. It also gives you a flavour of the academic and social life at Queen's.
If you cannot find the information you need here, please contact the University Admissions and Access Service (admissions@qub.ac.uk), giving full details of your qualifications and educational background.
International Students
Our country/region pages include information on entry requirements, tuition fees, scholarships, student profiles, upcoming events and contacts for your country/region. Use the dropdown list below for specific information for your country/region.
English Language Requirements
An IELTS score of 6.0 with a minimum of 5.5 in each test component or an equivalent acceptable qualification, details of which are available at: http://go.qub.ac.uk/EnglishLanguageReqs
If you need to improve your English language skills before you enter this degree programme, INTO Queen's University Belfast offers a range of English language courses. These intensive and flexible courses are designed to improve your English ability for admission to this degree.
Academic English: an intensive English language and study skills course for successful university study at degree level
Pre-sessional English: a short intensive academic English course for students starting a degree programme at Queen's University Belfast and who need to improve their English.
International Students - Foundation and International Year One Programmes
INTO Queen's offers a range of academic and English language programmes to help prepare international students for undergraduate study at Queen's University. You will learn from experienced teachers in a dedicated international study centre on campus, and will have full access to the University's world-class facilities.
These programmes are designed for international students who do not meet the required academic and English language requirements for direct entry.
Studying for a degree in Applied Mathematics and Physics at Queen’s will assist students in developing the core skills and employment-related experiences that are valued by employers, professional organisations and academic institutions. Graduates from this degree at Queen’s are well regarded by many employers (local, national and international) and over half of all graduate jobs are now open to graduates of any discipline, including mathematics.
Although many of our graduates are interested in pursuing careers in teaching, banking and finance, significant numbers develop careers in a wide range of other sectors. The following is a list of the major career sectors that have attracted our graduates in recent years:
Management Consultancy
Export Marketing (NI Programme)
Fast Stream Civil Service
Varied graduate programmes (Times Top 100 Graduate Recruiters/AGR, Association of Graduate Recruiters UK)
Other Career-related information:
Queen’s is a member of the Russell Group and, therefore, one of the 20 universities most-targeted by leading graduate employers. Queen’s students will be advised and guided about career choice and, through the Degree Plus initiative, will have an opportunity to seek accreditation for skills development and experience gained through the wide range of extra-curricular activities on offer. See Queen’s University Belfast’s Employability Statement for further information.
Degree Plus and other related initiatives:
Recognising student diversity, as well as promoting employability enhancements and other interests, is part of the developmental experience at Queen’s. Students are encouraged to plan and build their own, personal skill and experiential profile through a range of activities including; recognised Queen’s Certificates, placements and other work experiences (at home or overseas), Erasmus study options elsewhere in Europe, learning development opportunities and involvement in wider university life through activities, such as clubs, societies, and sports.
Queen’s actively encourages this type of activity by offering students an additional qualification, the Degree Plus Award (and the related Researcher Plus Award for PhD and MPhil students). Degree Plus accredits wider experiential and skill development gained through extra-curricular activities that promote the enhancement of academic, career management, personal and employability skills in a variety of contexts. As part of the Award, students are also trained on how to reflect on the experience(s) and make the link between academic achievement, extracurricular activities, transferable skills and graduate employment. Participating students will also be trained in how to reflect on their skills and experiences and can gain an understanding of how to articulate the significance of these to others, e.g. employers.
Overall, these initiatives, and Degree Plus in particular, reward the energy, drive, determination and enthusiasm shown by students engaging in activities over-and-above the requirements of their academic studies. These qualities are amongst those valued highly by graduate employers. http://www.qub.ac.uk/directorates/degreeplus/
What employers say
The Regional Medical Physics Service benefits hugely from the high quality graduates produced by the School of Maths and Physics at Queen’s University Belfast. Since the turn of the century, two thirds of all our Clinical Scientists obtained a Physics degree at Queen’s prior to joining the Service. The vast majority of these degrees were at integrated masters level or higher.
Prof. Alan Hounsell, Head of the Regional Medical Physics Service, Belfast Health and Social Care Trust
At Citi, we value diverse thinking and we encourage Maths students to join our Graduate Programmes each year. The Maths graduates that we have hired have been extremely successful in their careers within Citi, in particular our Software Engineering opportunities. They excel in their profession with us as they are able to transfer their analytical, numerical and problem solving skills to their day to day responsibilities.
Aislinn Wilson, Applications Development Senior Manager, Citi group
We have Mathematics graduates working across many parts of the business and they play a central role in creating cutting edge solutions for our customers, enabling them to push the boundaries of science.
Claire Greenwood, Director of Engineering, Andor Technology
At Randox we continuously pursue disruptive technologies to push the cutting edge of healthcare diagnostics. QUB Physics graduates are an excellent fit in this framework as at the core of their studies is a questioning of methodology and quantitative analysis.
Dr. Paul Vance, Project Manager, Randox Laboratories
Degree Plus/Future Ready Award for extra-curricular skills
In addition to your degree programme, at Queen's you can have the opportunity to gain wider life, academic and employability skills. For example, placements, voluntary work, clubs, societies, sports and lots more. So not only do you graduate with a degree recognised from a world leading university, you'll have practical national and international experience plus a wider exposure to life overall. We call this Degree Plus/Future Ready Award. It's what makes studying at Queen's University Belfast special.
1EU citizens in the EU Settlement Scheme, with settled status, will be charged the NI or GB tuition fee based on where they are ordinarily resident. Students who are ROI nationals resident in GB will be charged the GB fee.
2 EU students who are ROI nationals resident in ROI are eligible for NI tuition fees.
3 EU Other students (excludes Republic of Ireland nationals living in GB, NI or ROI) are charged tuition fees in line with international fees.
The tuition fees quoted above for NI and ROI are the 2024/25 fees and will be updated when the new fees are known. In addition, all tuition fees will be subject to an annual inflationary increase in each year of the course. Fees quoted relate to a single year of study unless explicitly stated otherwise.
Tuition fee rates are calculated based on a student’s tuition fee status and generally increase annually by inflation. How tuition fees are determined is set out in the Student Finance Framework.
Additional course costs
All essential software will be provided by the University, for use on University facilities, however for some software, students may choose to buy a version for home use.
All Students
Depending on the programme of study, there may be extra costs which are not covered by tuition fees, which students will need to consider when planning their studies.
Students can borrow books and access online learning resources from any Queen's library. If students wish to purchase recommended texts, rather than borrow them from the University Library, prices per text can range from £30 to £100. Students should also budget between £30 to £75 per year for photocopying, memory sticks and printing charges.
Students undertaking a period of work placement or study abroad, as either a compulsory or optional part of their programme, should be aware that they will have to fund additional travel and living costs.
If a programme includes a major project or dissertation, there may be costs associated with transport, accommodation and/or materials. The amount will depend on the project chosen. There may also be additional costs for printing and binding.
Students may wish to consider purchasing an electronic device; costs will vary depending on the specification of the model chosen.
There are also additional charges for graduation ceremonies, examination resits and library fines.
How do I fund my study?
There are different tuition fee and student financial support arrangements for students from Northern Ireland, those from England, Scotland and Wales (Great Britain), and those from the rest of the European Union.
Application for admission to full-time undergraduate and sandwich courses at the University should normally be made through the Universities and Colleges Admissions Service (UCAS). Full information can be obtained from the UCAS website at: www.ucas.com/students.
When to Apply
UCAS will start processing applications for entry in autumn 2025 from early September 2024.
The advisory closing date for the receipt of applications for entry in 2025 is still to be confirmed by UCAS but is normally in late January (18:00). This is the 'equal consideration' deadline for this course.
Applications from UK and EU (Republic of Ireland) students after this date are, in practice, considered by Queen’s for entry to this course throughout the remainder of the application cycle (30 June 2025) subject to the availability of places. If you apply for 2025 entry after this deadline, you will automatically be entered into Clearing.
Applications from International and EU (Other) students are normally considered by Queen's for entry to this course until 30 June 2025. If you apply for 2025 entry after this deadline, you will automatically be entered into Clearing.
Applicants are encouraged to apply as early as is consistent with having made a careful and considered choice of institutions and courses.
The Institution code name for Queen's is QBELF and the institution code is Q75.
Additional Information for International (non-EU) Students
Applying through UCAS
Most students make their applications through UCAS (Universities and Colleges Admissions Service) for full-time undergraduate degree programmes at Queen's. The UCAS application deadline for international students is 30 June 2025.
Applying direct
The Direct Entry Application form is to be used by international applicants who wish to apply directly, and only, to Queen's or who have been asked to provide information in advance of submitting a formal UCAS application. Find out more.
Applying through agents and partners
The University’s in-country representatives can assist you to submit a UCAS application or a direct application. Please consult the Agent List to find an agent in your country who will help you with your application to Queen’s University.
