Mathematical Physics MSci

The idea of proof is fundamental to all mathematics. We’ll look at mathematical reasoning using techniques from logic to deal with sets, functions, sequences and series.

This module links directly with your study in Calculus and Linear Mathematics. It provides you with the foundations for the broader area of Mathematical Analysis. This includes the rigorous study of the infinite and the infinitesimal.

You will also learn the basics of computer programming. This will give you the chance to use computational algorithms to explore many of the mathematical results you’ll encounter in your core modules.

Your study will include:

• propositional and predicate logic; set theory, countability
• proof: direct, indirect and induction
• sequences and infinite series (convergence and divergence)
• limits and continuity of functions
• programming in Python

How do we define calculus? How is it used in the modern world?

The concept can be explained as the mathematics of continuous change. It allows us to analyse motion and change in time and space.

You will cover techniques for differentiating, integrating and solving differential equations. You’ll learn about the theorems which prove why calculus works. We will explore the theory and how it can be applied in the real world.

Your study will include:

• functions: limits, continuity and differentiability, rules of differentiation
• techniques for integration, fundamental theorem of calculus
• solution of linear and nonlinear differential equations
• multivariate calculus, Lagrange multipliers, stationary points
• multiple integrals, changes of variables, Jacobians

This module gives you the mathematical tools required for later modules which involve modelling with differential equations. These include:

• mathematical physics
• mathematical medicine and biology
• scientific computation

Vectors, matrices and complex numbers are familiar topics from A level Mathematics and Further Mathematics. Their common feature is linearity. A linear mathematical operation is one which is compatible with addition and scaling.

As well as these topics you’ll study the concept of a vector space, which is fundamental to later study in abstract algebra. We will also investigate practical aspects, such as methods for solving linear systems of equations.

The module will give you the tools to analyse large systems of equations that arise in mathematical, statistical and computational models. For example, in areas such as:

• fluid and solid mechanics
• mathematical medicine and biology
• mathematical finance

Your study will include:

• complex numbers, vector algebra and geometry
• matrix algebra, inverses, determinants
• vector spaces, subspaces, bases
• linear systems of simultaneous equations, Gaussian elimination
• eigenvalues and eigenvectors, matrix diagonalisation
• linear transformations, inner product spaces

How does the world really work?

We’ll take you from Newton’s mechanics, the pinnacle of the scientific revolution and the foundation of our understanding of modern physics, right through to our current understanding of physics with Einstein’s theory of relativity and quantum mechanics.

This module will underpin your entire physics degree. It contains all the ideas and principles that form the basis of our modern world. As you’ll find out, some of these ideas are very strange indeed.

You’ll study:

• Newton’s laws of mechanics
• The physics of waves and oscillations
• Electricity and magnetism
• Quantum mechanics and the foundations of modern physics
• Einstein’s relativity

This year-long module will train you in the mathematical modelling of physical processes. You’ll cover topics such as basic statistics and errors, dimensional analysis, curve sketching, orders of magnitude and estimates, and integrating problems in physics among others.

You’ll receive training in basic computing techniques using Python, and will be introduced to their use in solving physical problems.

You’ll spend two hours in computer classes and a one hour lecture each week. 

This course explores the classical and quantum mechanical description of motion. The laws of classical mechanics are investigated both in their original formulation due to Newton and in the mathematically equivalent but more powerful formulations due to Lagrange and Hamilton. Applications are made to problems such as planetary motion, rigid body motion and vibrating systems. Quantum mechanics is developed in terms of a wave function obeying Schroedinger's equation, and the appropriate mathematical notions of Hermitian operators and probability densities are introduced. Applications include problems such as the harmonic oscillator and a particle in a three-dimensional central force field. 

This course aims to give students a sound grounding in the application of both differential and integral calculus to vectors, and to apply vector calculus methods and separation of variables to the solution of partial differential equations. The module is an important pre-requisite for a wide range of other courses in Applied Mathematics.

Macroscopic systems exhibit behaviour that often differs from that of their microscopic constituents. This module explores the relationship between the macro and micro worlds, and the complexity which emerges from the interplay of many interacting degrees of freedom.

You’ll study:

• Laws of thermodynamics, and how they are still relevant
• Macroscopic characterisation of matter, for example how liquid nitrogen is made and understood
• Statistical formulation, linking micro and macro systems
• Quantum statistics, providing a theory for everything!

This is a core module targeted at year 2 Mathematical Physics students and Natural Sciences students on the Maths and Physics pathway. You’ll study the physics of light and Maxwell’s equations on electrodynamics.

This is a core module targeted at year 2 Mathematical Physics students and Natural Sciences students on the Maths and Physics pathway. You’ll study:

• the physics of light, interference, diffraction, interferometry and the construction of optical instruments
• Maxwell’s equations on electrodynamics, and their applications in electrostatics, magnetic fields and electromagnetic waves.

In this module you will apply the general theory you learnt in Introduction to Mathematical Physics to more general problems. New topics will be introduced such as the quantum theory of the hydrogen atom and aspects of angular momentum such as spin.

This module will introduce students to the physics of atoms, nuclei and the fundamental constituents of matter and their interactions. The module will also develop the quantum mechanical description of these.

Topics to be covered are:

• Approximation techniques first order perturbation theory, degeneracies, second order perturbation theory, transition rates, time-dependent perturbation theory, Fermi's golden rule
• Particle Physics protons and neutrons, antiparticles, particle accelerators and scattering experiments, conservation laws, neutrinos, leptons, baryons and hadrons, the quark model and the strong interaction, weak interactions, standard model
• Introduction to atomic physics review of simple model of hydrogen atom, Fermi statistics and Pauli principle, aufbau principle, hydrogenic atoms, exchange, fine structure and hyperfine interactions, dipole interaction, selection rules and transition rates
• Lasers optical polarization and photons, optical cavities, population inversions, Bose statistics and stimulated emission, Einstein A and B coefficients
• Nuclear Physics Radioactivity, decay processes, alpha, beta and gamma emission, detectors, stability curves and binding energies, nuclear fission, fusion, liquid drop and shell models.

Solid state physics underpins almost every technological development around us, from solar cells and LEDs to silicon chips and mobile phones.

The aim of this module is to introduce to you the fundamental topics in solid state physics. We start by looking at why atoms and molecules come together to form a crystal structure. We then follow the electronic structure of these through to interesting electronic, thermal and magnetic properties that we can harness to make devices.

You’ll study:

• Why atoms and molecules come together to form crystal structures
• The description of crystal structures, reciprocal lattices, diffraction and Brillouin zones
• Nearly-free electron model – Bloch's theorem, band gaps from electron Bragg scattering and effective masses
• Band theory, Fermi surfaces, qualitative picture of transport, metals, insulators and semiconductors
• Semiconductors – doping, inhomogeneous semiconductors, basic description of pn junction
• Phonons normal modes of ionic lattice, quantization, Debye theory of heat capacities, acoustic and optical phonons
• Optical properties of solids absorption and reflection of light by metals, Brewster angle, dielectric constants, plasma oscillations
• Magnetism – Landau diamagnetism, paramagnetism, exchange interactions, Ferromagnetism, antiferromagnetism, neutron scattering, dipolar interactions and domain formation, magnetic technology

In this module you’ll have an introduction to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain new phenomena which occur in relativity.

You’ll carry out a project within the areas of chemical and molecular physics, which may be experimental or theoretical in nature.

Spending around two hours per week in lectures and tutorials, you’ll work in pairs to plan your project under the guidance of a project supervisor.

In this module you’ll systematically study black holes and their properties, including astrophysical processes, horizons and singularities. You’ll have an introduction to black hole radiation to give you an insight into problems of research interest. You’ll gain knowledge to help you begin research into general relativity.

In this module you’ll be equipped with the tools and knowledge to extend your understanding of general relativity. You’ll explore more abstract and powerful concepts using examples of curved space-times such as Lie groups and manifolds among others. 

This module gives you a mathematical introduction to quantum information theory. The aim is to provide you with a background in quantum information science which will facilitate further independent learning and allow you to understand the scope and nature of current research topics.

What is gravity? To Isaac Newton it was the force that made the apple fall, or that held the planets in orbit around the Sun. To a particle physicist, it is the exchange of virtual gravitons. To a string theorist, it is the exchange of closed strings. For us, in this course, gravity will be a fake! It isn’t a force at all. It is the shape of spacetime. Gravity is geometry, and for that you need to learn some funky mathematics. That is what this module is all about.

What is gravity? To Isaac Newton it was the force that made the apple fall, or that held the planets in orbit around the Sun. To a particle physicist, it is the exchange of virtual gravitons. To a string theorist, it is the exchange of closed strings. For us, in this course, gravity will be a fake! It isn’t a force at all. It is the shape of spacetime. Gravity is geometry, and for that you need to learn some funky mathematics. That is what this module is all about.

You’ll learn about:

• Spacetime manifolds – the fabric of gravitational physics
• Tensor calculus – the changing dynamics of physical objects as they weave their way through space and time
• Curvature – what it really means for spacetime to be curved and why this has anything to do with gravity
• Einstein’s equations – arguably the best equations in all of physics
• Solving the best equations in all of physics (spoiler: it’s generally super hard, but we do have one or two tricks)
• Extra dimensions and black holes – all the cool stuff basically.

Particle physics has been hugely influential in both science and society, from the discovery of the electron to the detection of the Higgs boson. In this module you will be introduced to the mathematical tools required to understand our current description of the Standard Model of particle physics.

You’ll study:

• The Dirac equation, which describes electrons, quarks and neutrinos
• How symmetry and conservation laws are crucial in particle physics
• The Feynman approach to computing the scattering of particles

• Bose condensation review of Bose statistics, BEC, BEC in cold atomic gases.
• Superfluidity in Helium-4 quantum fluids, macroscopic wave functions, superfluidity, non-classical rotational inertia and vortices, phonon and roton excitations.
• Superconductivity conduction in metals, superconducting materials, zero-resistivity, Meissner effect, perfect diamagnetism, type I and type II behaviour, London theory.
• BCS theory of superconductivity.- electron-phonon interaction, Cooper pairs, BCS wave function, order parameter and microscopic origin of GL.
• Applications: squids, superconducting magnets etc.

In this year-long module you’ll be introduced to the study of the quantum dynamics of relativistic particles. You’ll learn about the quantum description of electrons, photons and other elementary particles leading to an understanding of the standard model of particle physics.