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 Theoretical Physics with Mathematics MSci Hons (F3G1)
Theoretical Physics with Mathematics MSci Hons  2019 Entry
UCAS Code
F3G1
Entry Year
2019
also available in 2018
A Level Requirements
AAA
see all requirements
see all requirements
Duration
Full time 4 Year(s)
Course Overview
Taught jointly with Lancaster’s Department of Mathematics and Statistics, our degree in Theoretical Physics with Mathematics combines core physics and specialised theoretical physics subjects with classes in pure mathematics. This equips you with an understanding of the mathematical foundations of physics, for example, you will learn how quantum mechanics is underpinned by the powerful mathematical concept of a Hilbert space.
In your first year you will cover the core of physics in modules such as Quantum Physics and Electromagnetism, and the core of mathematics including geometry and calculus, numbers and relations, and probability.
In years two and three, the core physics modules are complemented by courses from the Theoretical Physics scheme and mathematical topics such as group theory and differential equations. In the final year of your course you will take advanced options such as Quantum Information and Advanced Gravity and Relativity, and complete your extended research project on a topic such as Gravitational Waves, Quantum Computation, Physics of Graphene, Photonic Crystals, and Geometry and Electrodynamics.
We offer you a range of possible topics for your extended final year project. Recent project topics include adatoms on graphene, silicine, photonic crystals, topological insulators, fundamentals of radiation reaction and gravitational waves.
Entry Requirements
Grade Requirements
A Level AAA
Required Subjects A level Mathematics grade A and A level Physics grade A
IELTS 6.0 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Interviews Applicants may be interviewed before being made an offer.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL and Physics HL
BTEC May be considered alongside A level Mathematics and A level Physics.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Classical Mechanics
In Classical Mechanics students will apply the ideas of fundamental Newtonian mechanics to real largescale systems such as rotating bodies, planetary systems and classical fluids.
The focus is on gravitation and its central importance in determining the largescale behaviour of the Universe. Concepts such as inertial and gravitational mass, Mach's principle, black holes and even dark matter will be explored.
This module will consider how to extend the principles of basic kinematics and dynamics to rotational situations, giving students an understanding of concepts of torque, moment of inertia, centre of mass, angular momentum and equilibrium.
Time will be spent looking at how to describe basic processes in the properties of materials including elasticity of solids and fluid dynamics.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Electric and Magnetic Fields
Covering the basic laws of electromagnetism, this module allows students to investigate the similarities and differences between electric and magnetic fields, and to explore the basic concepts of electromagnetic phenomena including charge, current, field, force and potential.
The module will begin by studying electrostatics, describing forces and fields due to charge distributions using Coulomb's law and Gauss's law. Students will also look at the concept of polarisation, and how this can be applied to capacitance and combinations of capacitors.
Later on, the module will introduce magnetostatics, and students will learn how to describe it using the concepts of field, flux and force, and the motion of charged particles in a magnetic field. They will also look at the origins of magnetic fields and Ampere's law, and Faraday's law of electromagnetic induction.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. They will see how partial derivatives can help to understand surfaces, while repeated integrals enable them to calculate volumes. The module will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, rates of change are introduced with respect to several quantities. How to find maxima and minima will be explained. Applications include the method of least squares. Finally, various methods for solving differential equations of one variable will be investigated.

Integration and Differentiation
Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.
The student will learn how to express a linear transformation of the real euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.

Numbers and Relations
The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Quantum Physics
The ultimate description of the Universe requires quantum and not classical mechanics. This module begin by investigating how specific experiments led to the breakdown of classical physics, before moving into the quantum world.
Students will look at the basic ideas of wave mechanics, particularly wave particle duality, as well as considering the probabilistic nature of phenomena and the uncertainty principle through the Schrodinger equation and its solution for simple situations.
Other topics that will be studied include the photoelectric effect, the nuclear atom and single slit diffraction. Ultimately, the students will be able to apply their knowledge to modelling real phenomena and situations.

Statistics
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.

The Physical Universe
In this module students will have the opportunity to explore the nature and methods of physics by considering the different scales of the Universe and the areas of physics which relate to them.
They will model real phenomena and situations, looking at the physical principles which are fundamental to mechanics, particularly Newton’s laws relating to forces and motion, and the principles of the conservation of energy and momentum.
Later on, the Special Theory of Relativity will be the main focus, beginning with Einstein's postulates and moving on to inertial reference frames, the physics of simultaneity, length contraction and time dilation, and spacetime diagrams.

Thermal Properties of Matter
This module focuses on the study of the thermal properties of matter, during which the students will gain an understanding of how to relate them to the fundamental mechanical properties of systems.
It will begin with an introduction to the concepts of temperature and heat, thermal equilibrium and temperature scales. Then students will look at how to describe mechanisms of heat transfer, particularly in phase changes and equations of state, and the kinetic model of an ideal gas.
As part of the module students will also have the opportunity to explore the first and second laws of thermodynamics, including concepts of internal energy, heat and work done, heat engines and refrigerators, and entropy. They will then learn about the role of thermodynamics in describing macroscopic physical situations, looking in particular at temperature, entropy, work, heat, and internal energy.

Vector Calculus
This module is ideal for students looking to develop their understanding of vector algebra and coordinate geometry in a physical context, extending elementary ideas of functions and calculus to a threedimensional description based on vector fields and potentials.
You’ll begin by exploring the real functions of many variables and their partial derivatives, followed by implicit differentiation of the functions of many variables and the chain rule. You’ll then go on to study the gradient vector in three dimensions in relation to directional derivatives, and will investigate the divergence and curl of a vector field as well as Stokes' theorem and the divergence theorem.
Vector Calculus places a focus on calculus in higher dimensional space, allowing you to develop your knowledge of parametric representations of curves, surfaces and volumes, calculation of areas and volumes including the use of changes of variables and Jacobians, and the calculation of line and surface integrals.
Core
Year 2

Abstract Algebra
This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here,students will select a small number of properties which these and other examples have in common, and use them to define a group.
They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure',a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a wayto abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.

Complex Analysis
Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.
The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every nonconstant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Electromagnetism, Waves and Optics
This module provides students with a working knowledge of electromagnetion through Maxwell’s equations using the tools of vector calculus. Students will become familiar with the common connections between the many different phenomena in nature that share the mathematical model of a harmonic oscillator or of a wave. This module addresses the basic properties of wave propagation, diffraction and inference, and laser operation.
Students will develop an appreciation for the power of vector calculus and Maxwell’s equations for the description of electromagnetic phenomena, and will gain practical knowledge of Fresnel and Fraunhofer diffraction, as well as thinfilm interference fringes and antireflection coatings. Additionally, the module aims to enhance students’ understanding of the origin of polarisation, and the relevance of dichroism, along with an understanding of the basic elements of a laser, laser operation and important features of laser light.

Experimental Physics, Skills and Mechanics
The module expands students’ knowledge on topics such as Newton's laws, solution of onedimensional dynamical problems, and Lagrangian, its relation to Newton's equations and the least action principle. This module includes lectures on analytical methods used both in classical mechanics and in broader areas of theoretical and mathematical physics. Students are given the opportunity to perform experiments in optics, mechanics and electric circuits which illustrate and compliment the taught material, and are additionally required to write a scientific report on one of the experiments.
By the end of the module, students will understand methods of integration of equations of motion for dynamical problems in classical mechanics, be able to use variational calculus in application to functionals and exploit the generality of Lagrangian and Hamiltonian techniques by using appropriate generalised coordinates. They will also be acquainted with the concepts of phase space, stability of motion and chaos. Additionally, students will develop useful techniques for experimental data collection and analysis and understand how to assess the statistical validity of data and their interpretation.

Linear Algebra II
Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.

Quantum Mechanics
Students will be introduced to various axioms for quantum mechanics, such as eigenvalues, diagonalisation, differential and matrix operators and commutation relations. They will also learn about rotations and angular momentum, the interaction of magnetic moment with static magnetic field and electron spin. Students can expect to investigate approximation methods, such as the timedependent RayleighSchrodinger perturbation theory, and time dependent interactions, including the Heisenberg picture and time dependent Hamiltonians.
Students will learn to apply quantum mechanics to problems in one and three dimensions, including the hydrogen atom, by solving the Schrödinger equation, and will develop the ability to find approximate solutions for not exactly solvable systems. The module will enhance students’ understanding of expectation values and probabilities in the context of experiments on quantum systems, along with an appreciation for the mathematical consistency of quantum mechanics.

Real Analysis
A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

Relativity, Nuclei and Particles
Students receive an introductory conceptsbased approach to the module, giving a basic understanding of nuclei and fundamental particles. The module covers the general properties of nuclei, such as composition, the forces within the nucleus, mass and binding energy. Students are then introduced to the standard model of particle physics, including the three generations of fundamental particles.
By the end of the module, students will gain a working knowledge of Einstein’s theory of special relativity, both conceptually and mathematically, and will understand why the theory has replaced Newton’s concepts of absolute space and time. Additionally, students will develop a broad understanding of the equivalence principle and its relevance for general relativity.
Core
Year 3

Atomic Physics
This module introduces oneelectron atoms and the spinorbit magnetic interaction, along with identical particles and the Helium atom. Students will investigate the Fermi gas model and the single particle shell model, and will compare predictions of the shell model for nuclear spins, parities and magnetic moments with experimental results. The module explores the nuclear beta decay process and the Fermi and Gamow  Teller selection rules, and students are provided with a description of the beta decay rate and the electron energy spectrum in terms of a nuclear matrix element and a statistical factor.
Students will develop their knowledge in atomic and nuclear physics to an advanced level, and will be able to use the results of basic quantum mechanics to explain the basic characteristics of atomic and nuclear structure, in addition to gaining the ability to describe the processes of atomic transitions and nuclear decays. The module will provide an explanation of the concept and importance of the parity of an atomic or nuclear state, and will provide students with the opportunity to study the nuclear beta decay process and in particular the neutrino and parity nonconservation.

Particle Physics
The module explores symmetries, the Quark model and gives an introduction to QCD. Students will explore leptons, as well as forces and their carrier particles and Feynman diagrams. The module aims to provide a general introduction to theoretical and experimental topics in elementary particle physics, essentially the Standard Model of particle physics.
Students will gain the ability to describe the main features of the Standard Model of particle physics and understand its place in physics as a whole, and will be able to identify major pieces of experimental evidence supporting the key theoretical ideas, including the experimental techniques used, such as accelerators and detectors. In addition, students will understand the role of symmetry and conservation laws in fundamental physics, and will develop the ability to perform calculations of physically observable quantities relevant to the subject, along with solving problems based on the application of the general principles of particle physics.

Solid State Physics
The module offers an introduction to reciprocal lattice and diffraction of waves, the electronic band structure in metals, and insulators and semiconductors. Students will explore electrons in semiconductors, effective mass and the heat capacity of solids. There will be a Summary of experimental phenomena, tunnelling, Josephson Junctions and an outline of BCS theory.
Students will be introduced to theoretical and experimental topics in solid state physics at an advanced level, and will develop an understanding of the main features of the physics of electrons in solids, along with knowledge of the main features of the optical properties of solids.
Students will gain an enhanced understanding of crystal lattices and phonons, along with the main features of the thermal properties of solids, and will be able to describe major pieces of experimental evidence supporting the key theoretical ideas, including the experimental techniques used.

Statistical Physics
This module explores the ideas, techniques and results of statistical physics. Students will examine gases and the density of states, along with the statistics of gases, fermions and bosons and the two distributions for gases. MaxwellBoltzmann gases, velocity distribution and fermiDirac gases are investigated as the module provides an uncomplicated and direct approach to the subject, using frequent illustrations from low temperature physics.
Students will provide a unified survey of the statistical physics of gases, including a full treatment of quantum statistics, gaining a fuller insight into the meaning of entropy. Students will gain knowledge in applications of statistics to various types of gas. Ultimately, students will develop the ability to apply expressions and distributions in order to form accurate deductions, for example using the Boltzmann distribution for the probability of finding a system in a particular quantum state. Additionally, students will learn the role of statistical concepts in understanding macroscopic systems, and will be able to describe superfluidity in liquid helium, BoseEinstein condensation and black body radiation.

Theoretical Physics Group Project
In the Theoretical Physics Group Project, students will work as part of a team and will receive guidance on project management, planning and meetings. The module will culminate in a writingup stage in which the groups will prepare a group report, and students will be presented an opportunity to give an individual talk at the physics miniconference.
The module equips students with the ability to develop a theoretical physics research project with formulation, literature searches, data gathering, analysis and presentation.

Theoretical Physics with Mathematics Independent Study
This module requires independent study in various aspects of theoretical physics is guided by a series of workshops. Students will solve problems, partake in intense reading, and prepare and deliver presentations. Students will be provided with an opportunity to extend their preliminary studies by undertaking openended investigations into various aspects and problems of theoretical physics, writing up their findings in an individual report.
The module will teach analytical recipes of theoretical physics used in quantum mechanics, with the focus on the variational functions method, operator techniques with applications in perturbation theory methods and coherent states of a quantum harmonic oscillator, whilst students will train in the use of the operator algebra of 'creation' and 'annihilation' operators in the harmonic oscillator problem, which will develop a basis for the introduction of second quantisation in manybody systems. Students will develop an understanding in the algebra of creation and annihilation operators for Bose and Fermi systems, along with exploring the secondquantised representation of Hamiltonians of interacting manybody systems.
Additionally, students will analyse BoseEinstein condensation in one, two, and threedimensional systems and will develop the ability to describe the condensate using the method of coherent states, as well as covering the GinzburgLandau theory of a superfluid phase transition and to describe vortices in a superfluid. They will learn to relate Bose and Fermi statistics to the symmetry of manybody systems with respect to permutations of identical particles and will receive computer programming skills, as well as gaining experience in individual project work. The module’s main aim is to prepare students to enable them to undertake fourth year theoretical physics projects.
Core

Differential Equations
Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.
While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.

Elliptic Curves
Students will be given a solid foundation in the basicsof algebraic geometry. They will explore how curves can be described by algebraic equations, and learn how to use abstract groups in dealing with geometrical objects (curves). The module will present applications and results of the theory of elliptic curves and provide a useful link between concepts from algebra and geometry.
Students will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally, they will learn to perform algebraic computations with elliptic curves.

Geometry of Curves and Surfaces
The topic of smooth curves and surfaces in threedimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.
A number of wellknown concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

Groups and Symmetry
Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.
Situations will be considered in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.
Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.

Hilbert Space
Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.
Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.

Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.

Linear Systems
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a (A,B,C,D) system.

Metric Spaces
An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.
They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

Number Theory
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the nonspecialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Representation Theory of Finite Groups
Students will cover the basics of ordinary representation theory. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between the two will be discussed.
The second part is an introduction to the ordinary character theory of finite groups, intrinsic to representation theory. Students will learn the concepts of Rmodule and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups.
They will also learn to perform computations with representations and morphisms in a selection of finite groups

Rings Fields and Polynomials
Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.
They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
How wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.
Optional
Year 4

MPhys Literature Review
Information for this module is currently unavailable.

MPhys Project
MPhys projects vary from year to year and are tailored to suit the individual student and the available research facilities. This twomodule project commences with a dissertation or literature review. Students will write a report on the project work and will conduct a presentation for the miniconference in the summer term, along with gaining skills related to oral presentation of scientific research.
Project work gives students the opportunity to carry out research or a detailed investigation into a specific area of physics appropriate to their chosen degree theme. Students will develop and apply analytical and problemsolving skills in an open ended situation, involving use of the library, computer, and other resources as appropriate, working alone or in a small group.
By the end of this module, students will demonstrate the ability to plan, manage and execute an investigation an area of physics in a systematic way using appropriate techniques. They will formulate conclusions and critically compare with relevant theory, and may be required to generate and analyse data and critically assess experimental uncertainties.
Core

Advanced Electrodynamics and Gravity
This module introduces a range of geometry and exterior calculus, including scalar fields, vector fields and convector fields. Students will explore pforms, exterior derivative, metrics and Hodge dual, and will discover electrodynamics, more specifically Maxwell equations in terms of the Maxwell 2form, 4velocity fields and Lorentz force equation in terms of the Maxwell 2form. Gravity is also covered, and students will engage in topics such as Einstein 3forms, stressenergymomentum 3forms and Einstein equations. Additionally, students will gain knowledge of killing vectors, spacetimes with symmetry, conserved quantities and black holes.
Students will gain the knowledge required to display an understanding of the intrinsic, covariant nature of electrodynamic, along with a familiarity with handling the Einstein equations and field equations on curved spacetime. Students will also be able to formulate and tackle field theories on spacetime using tools from modern differential geometry.

Advanced Magnetism
Students will be offered a revision of elements of the theory of electromagnetism, before being introduced to the phenomenology of solid state magnetic phenomena. The module discusses Van Vleck's description of diamagnetism and diamagnetism as quantum phenomenon. Students will explore ferromagnetism and antiferromagnetism, ferromagnetic exchange and the Heisenberg model, which includes selfconsistent mean field theory. A description of ferromagnetic phase transitions and Curie temperature will be provided as part of the module, along with the elements of the GinzburgLandau theory of magnetic phase transitions.
By the end of the module, students will develop a knowledge and understanding of magnetic and electric phenomena in condensed matter physics, in addition to an enhanced awareness of recent advances and current problems in condensed matter physics.

Advanced Relativity and Gravity
The module offers a short review of special relativity, tensor calculus on Minkowski spacetime, differential calculus on Minkowski spacetime, and curved spacetimes. Students will explore general relativity, gravity as intrinsic curvature of spacetime, and the Einstein equations, along with predictions of the linearized Einstein equations, gravitational waves, and gravitomagnetic field equations. Students will investigate exact solutions of the Einstein equations, black holes and event horizons.
By the end of the module, students will have a basis knowledge and understanding of the theories of special and general relativity, and possess a conceptual understanding of the links between Newtonian mechanics and relativity. The module also provides a geometrical insight into the properties of spacetime and relativity.

Elliptic Curves
This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.
Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

Galois Theory
Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1to1 correspondence with the subgroups of G.
In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

Hilbert Space
Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.
The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

Lasers and Applications
This module will address the necessary requirements for laser action, spontaneous and stimulated emission rates, Einstein coefficients, optical gain coefficient, and characteristics of the emitted light. Students will become aware of the different types of lasers, such as gas and solid state, semiconductor, dye, chemical and excimer lasers. Semiconductor lasers: homojunction, single and double heterojunction devices will be investigated, along with materials and operating requirements. The module explores fabrication methods, quantum well lasers, advantages and characteristics. There will be a focus on a range of applications including laser surgery, optical fibre communications, laser machining, pollution monitoring and remote sensing.
By the end of the module, students will be familiarised with lasers and their applications, including the operating principles of a variety of different lasers. Students will understand the many uses of lasers in industry, medicine and the environment.

Lebesgue Integration
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Lie Groups and Lie Algebras
The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.
The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

Matter at low temperature
The module begins by discussing what physicists mean by high and low temperatures, and looks at the different types of ordering that may occur as systems cool. Students will explore cryogenic techniques used for accessing such low temperatures are described, including the design of useful cryostats. Students will observe the new phenomena that occur when systems are cooled below room temperature and will consider electron pairing leading to the zero resistance of superconducting materials, the effect of magnetic fields, and the role of macroscopic quantum mechanical wave functions. The module provides an overview of the practical uses in superconducting quantum interference devices (SQUIDs).
The module seeks to explore a selection of fascinating phenomena that occurs when cooling matter to temperatures more than a million times colder than the familiar 290K of everyday life and observe the significance for both physics and technology. Additionally, students will appreciate the relation between temperature and order, will know how low temperatures are produced, including dilution refrigerators, and will also be able to describe the phenomena of superconductivity and superfluidity.

Operator Theory
Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems. Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.
First, students will review Hilbert spaces, before spending some time studying infinitedimensional operators, notably the unilateral shift and multiplication operators, aswell as basic concepts. They will then consider the criteria for invertibilityof selfadjoint operators, leading to the spectral theory of such operators.

Probability Theory
The aim of this module is to develop an analytical and axiomatic approach to the theory of probabilities.
Students will consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. They will then use random variables and the expectation to develop a probability calculus, which is applied to achieve laws of large numbers for sums of independent random variables. Finally the characteristic function will be used to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.

Quantum transport in Low Dimensional Nanostructures
Students can expect to explore twodimensional electron systems, quantum transport in disordered lowdimensional electron systems and semiconductor quantum wires. The LandauerButtiker conductance formula is investigated, focusing on impurities in quantum wires, electronic transport in a magnetic field and the Hall effect. The module considers metallic point contacts, the pointcontact spectroscopy of the electronphonon interaction, and atomic breakjunctions and the scanning tunnelling microscope. Students will receive examples of applications of scanning tunnelling microscopy as part of the module.
By the end of the module, students will have knowledge of the physics of nanoscale solid state devices and how these may be manufactured and utilised. They will enhance their awareness of the recent advancements and current problems in condensed matter physics.

Representation Theory of Finite Groups
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of Rmodule and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.
The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.
The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

Semiconductor Device Physics
Students will familiarise themselves with crystal growth, including growth theory, faceting, impurity segregation and zone refining. The module presents students with a silicon case study, investigating semiconducting properties, silicon oxide, masking, surface pacification and photolithographic processing. Compound semiconductors will be discussed, covering band structure advantages over silicon, IIVI materials and effects of iconicity.
Additionally, students will explore thin film semiconductors, such as epitaxy, vapour phase growth, metalloorganic methods and liquid phase epitaxy, and the module provides a broad interdisciplinary overview of the linkage between the physics, chemistry and other materials sciences involved in the synthesis of semiconductors and the devices made from them.
By the end of the module, students will develop an understanding of the basic properties of crystals and crystal defects, and will be able to describe how crystals are grown and discuss the main semiconductor used for microelectronics as a detailed case study. Students will also demonstrate how physics continues to play a major role in enabling information technology.

Topology and Fractals
Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.
This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.
Careers
Careers
Physics is an exciting subject that is fundamental to the developments in modern society. Applications of the subject range from the very pure to the very practical, and a physics degree opens up a wide range of rewarding careers in scientific research and technological development, as well as in a variety of other professions. A substantial number of our graduates continue on to postgraduate education, or enter employment that directly relies on their specialist skills. Our students also find employment in careers where they are valued because of general skills gained during the course such as logical thinking, problem solving, numeracy and computer literacy. Examples include consulting, finance, computer programming, and accountancy, as well as managerial and administrative positions.
Lancaster University is dedicated to ensuring you not only gain a highly reputable degree, you also graduate with the relevant life and work based skills. We are unique in that every student is eligible to participate in The Lancaster Award which offers you the opportunity to complete key activities such as work experience, employability/career development, campus community and social development. Visit our Employability section for full details.
Fees and Funding
Fees
We set our fees on an annual basis and the 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU  Overseas 

£9,250  £18,890 
Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students taking Part I of the course are charged £10 for a bound copy of the lab manual. There are a number of optional oneday visits to places of interest and students pay travel costs.
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.

Course Overview
Course Overview
Taught jointly with Lancaster’s Department of Mathematics and Statistics, our degree in Theoretical Physics with Mathematics combines core physics and specialised theoretical physics subjects with classes in pure mathematics. This equips you with an understanding of the mathematical foundations of physics, for example, you will learn how quantum mechanics is underpinned by the powerful mathematical concept of a Hilbert space.
In your first year you will cover the core of physics in modules such as Quantum Physics and Electromagnetism, and the core of mathematics including geometry and calculus, numbers and relations, and probability.
In years two and three, the core physics modules are complemented by courses from the Theoretical Physics scheme and mathematical topics such as group theory and differential equations. In the final year of your course you will take advanced options such as Quantum Information and Advanced Gravity and Relativity, and complete your extended research project on a topic such as Gravitational Waves, Quantum Computation, Physics of Graphene, Photonic Crystals, and Geometry and Electrodynamics.
We offer you a range of possible topics for your extended final year project. Recent project topics include adatoms on graphene, silicine, photonic crystals, topological insulators, fundamentals of radiation reaction and gravitational waves.

Entry Requirements
Entry Requirements
Grade Requirements
A Level AAA
Required Subjects A level Mathematics grade A and A level Physics grade A
IELTS 6.0 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Interviews Applicants may be interviewed before being made an offer.
Other Qualifications
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL and Physics HL
BTEC May be considered alongside A level Mathematics and A level Physics.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk

Course Structure
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Classical Mechanics
In Classical Mechanics students will apply the ideas of fundamental Newtonian mechanics to real largescale systems such as rotating bodies, planetary systems and classical fluids.
The focus is on gravitation and its central importance in determining the largescale behaviour of the Universe. Concepts such as inertial and gravitational mass, Mach's principle, black holes and even dark matter will be explored.
This module will consider how to extend the principles of basic kinematics and dynamics to rotational situations, giving students an understanding of concepts of torque, moment of inertia, centre of mass, angular momentum and equilibrium.
Time will be spent looking at how to describe basic processes in the properties of materials including elasticity of solids and fluid dynamics.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Electric and Magnetic Fields
Covering the basic laws of electromagnetism, this module allows students to investigate the similarities and differences between electric and magnetic fields, and to explore the basic concepts of electromagnetic phenomena including charge, current, field, force and potential.
The module will begin by studying electrostatics, describing forces and fields due to charge distributions using Coulomb's law and Gauss's law. Students will also look at the concept of polarisation, and how this can be applied to capacitance and combinations of capacitors.
Later on, the module will introduce magnetostatics, and students will learn how to describe it using the concepts of field, flux and force, and the motion of charged particles in a magnetic field. They will also look at the origins of magnetic fields and Ampere's law, and Faraday's law of electromagnetic induction.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. They will see how partial derivatives can help to understand surfaces, while repeated integrals enable them to calculate volumes. The module will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, rates of change are introduced with respect to several quantities. How to find maxima and minima will be explained. Applications include the method of least squares. Finally, various methods for solving differential equations of one variable will be investigated.

Integration and Differentiation
Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.
The student will learn how to express a linear transformation of the real euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.

Numbers and Relations
The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Quantum Physics
The ultimate description of the Universe requires quantum and not classical mechanics. This module begin by investigating how specific experiments led to the breakdown of classical physics, before moving into the quantum world.
Students will look at the basic ideas of wave mechanics, particularly wave particle duality, as well as considering the probabilistic nature of phenomena and the uncertainty principle through the Schrodinger equation and its solution for simple situations.
Other topics that will be studied include the photoelectric effect, the nuclear atom and single slit diffraction. Ultimately, the students will be able to apply their knowledge to modelling real phenomena and situations.

Statistics
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.

The Physical Universe
In this module students will have the opportunity to explore the nature and methods of physics by considering the different scales of the Universe and the areas of physics which relate to them.
They will model real phenomena and situations, looking at the physical principles which are fundamental to mechanics, particularly Newton’s laws relating to forces and motion, and the principles of the conservation of energy and momentum.
Later on, the Special Theory of Relativity will be the main focus, beginning with Einstein's postulates and moving on to inertial reference frames, the physics of simultaneity, length contraction and time dilation, and spacetime diagrams.

Thermal Properties of Matter
This module focuses on the study of the thermal properties of matter, during which the students will gain an understanding of how to relate them to the fundamental mechanical properties of systems.
It will begin with an introduction to the concepts of temperature and heat, thermal equilibrium and temperature scales. Then students will look at how to describe mechanisms of heat transfer, particularly in phase changes and equations of state, and the kinetic model of an ideal gas.
As part of the module students will also have the opportunity to explore the first and second laws of thermodynamics, including concepts of internal energy, heat and work done, heat engines and refrigerators, and entropy. They will then learn about the role of thermodynamics in describing macroscopic physical situations, looking in particular at temperature, entropy, work, heat, and internal energy.

Vector Calculus
This module is ideal for students looking to develop their understanding of vector algebra and coordinate geometry in a physical context, extending elementary ideas of functions and calculus to a threedimensional description based on vector fields and potentials.
You’ll begin by exploring the real functions of many variables and their partial derivatives, followed by implicit differentiation of the functions of many variables and the chain rule. You’ll then go on to study the gradient vector in three dimensions in relation to directional derivatives, and will investigate the divergence and curl of a vector field as well as Stokes' theorem and the divergence theorem.
Vector Calculus places a focus on calculus in higher dimensional space, allowing you to develop your knowledge of parametric representations of curves, surfaces and volumes, calculation of areas and volumes including the use of changes of variables and Jacobians, and the calculation of line and surface integrals.
Core
Year 2

Abstract Algebra
This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here,students will select a small number of properties which these and other examples have in common, and use them to define a group.
They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure',a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a wayto abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.

Complex Analysis
Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.
The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every nonconstant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Electromagnetism, Waves and Optics
This module provides students with a working knowledge of electromagnetion through Maxwell’s equations using the tools of vector calculus. Students will become familiar with the common connections between the many different phenomena in nature that share the mathematical model of a harmonic oscillator or of a wave. This module addresses the basic properties of wave propagation, diffraction and inference, and laser operation.
Students will develop an appreciation for the power of vector calculus and Maxwell’s equations for the description of electromagnetic phenomena, and will gain practical knowledge of Fresnel and Fraunhofer diffraction, as well as thinfilm interference fringes and antireflection coatings. Additionally, the module aims to enhance students’ understanding of the origin of polarisation, and the relevance of dichroism, along with an understanding of the basic elements of a laser, laser operation and important features of laser light.

Experimental Physics, Skills and Mechanics
The module expands students’ knowledge on topics such as Newton's laws, solution of onedimensional dynamical problems, and Lagrangian, its relation to Newton's equations and the least action principle. This module includes lectures on analytical methods used both in classical mechanics and in broader areas of theoretical and mathematical physics. Students are given the opportunity to perform experiments in optics, mechanics and electric circuits which illustrate and compliment the taught material, and are additionally required to write a scientific report on one of the experiments.
By the end of the module, students will understand methods of integration of equations of motion for dynamical problems in classical mechanics, be able to use variational calculus in application to functionals and exploit the generality of Lagrangian and Hamiltonian techniques by using appropriate generalised coordinates. They will also be acquainted with the concepts of phase space, stability of motion and chaos. Additionally, students will develop useful techniques for experimental data collection and analysis and understand how to assess the statistical validity of data and their interpretation.

Linear Algebra II
Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.

Quantum Mechanics
Students will be introduced to various axioms for quantum mechanics, such as eigenvalues, diagonalisation, differential and matrix operators and commutation relations. They will also learn about rotations and angular momentum, the interaction of magnetic moment with static magnetic field and electron spin. Students can expect to investigate approximation methods, such as the timedependent RayleighSchrodinger perturbation theory, and time dependent interactions, including the Heisenberg picture and time dependent Hamiltonians.
Students will learn to apply quantum mechanics to problems in one and three dimensions, including the hydrogen atom, by solving the Schrödinger equation, and will develop the ability to find approximate solutions for not exactly solvable systems. The module will enhance students’ understanding of expectation values and probabilities in the context of experiments on quantum systems, along with an appreciation for the mathematical consistency of quantum mechanics.

Real Analysis
A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.

Relativity, Nuclei and Particles
Students receive an introductory conceptsbased approach to the module, giving a basic understanding of nuclei and fundamental particles. The module covers the general properties of nuclei, such as composition, the forces within the nucleus, mass and binding energy. Students are then introduced to the standard model of particle physics, including the three generations of fundamental particles.
By the end of the module, students will gain a working knowledge of Einstein’s theory of special relativity, both conceptually and mathematically, and will understand why the theory has replaced Newton’s concepts of absolute space and time. Additionally, students will develop a broad understanding of the equivalence principle and its relevance for general relativity.
Core
Year 3

Atomic Physics
This module introduces oneelectron atoms and the spinorbit magnetic interaction, along with identical particles and the Helium atom. Students will investigate the Fermi gas model and the single particle shell model, and will compare predictions of the shell model for nuclear spins, parities and magnetic moments with experimental results. The module explores the nuclear beta decay process and the Fermi and Gamow  Teller selection rules, and students are provided with a description of the beta decay rate and the electron energy spectrum in terms of a nuclear matrix element and a statistical factor.
Students will develop their knowledge in atomic and nuclear physics to an advanced level, and will be able to use the results of basic quantum mechanics to explain the basic characteristics of atomic and nuclear structure, in addition to gaining the ability to describe the processes of atomic transitions and nuclear decays. The module will provide an explanation of the concept and importance of the parity of an atomic or nuclear state, and will provide students with the opportunity to study the nuclear beta decay process and in particular the neutrino and parity nonconservation.

Particle Physics
The module explores symmetries, the Quark model and gives an introduction to QCD. Students will explore leptons, as well as forces and their carrier particles and Feynman diagrams. The module aims to provide a general introduction to theoretical and experimental topics in elementary particle physics, essentially the Standard Model of particle physics.
Students will gain the ability to describe the main features of the Standard Model of particle physics and understand its place in physics as a whole, and will be able to identify major pieces of experimental evidence supporting the key theoretical ideas, including the experimental techniques used, such as accelerators and detectors. In addition, students will understand the role of symmetry and conservation laws in fundamental physics, and will develop the ability to perform calculations of physically observable quantities relevant to the subject, along with solving problems based on the application of the general principles of particle physics.

Solid State Physics
The module offers an introduction to reciprocal lattice and diffraction of waves, the electronic band structure in metals, and insulators and semiconductors. Students will explore electrons in semiconductors, effective mass and the heat capacity of solids. There will be a Summary of experimental phenomena, tunnelling, Josephson Junctions and an outline of BCS theory.
Students will be introduced to theoretical and experimental topics in solid state physics at an advanced level, and will develop an understanding of the main features of the physics of electrons in solids, along with knowledge of the main features of the optical properties of solids.
Students will gain an enhanced understanding of crystal lattices and phonons, along with the main features of the thermal properties of solids, and will be able to describe major pieces of experimental evidence supporting the key theoretical ideas, including the experimental techniques used.

Statistical Physics
This module explores the ideas, techniques and results of statistical physics. Students will examine gases and the density of states, along with the statistics of gases, fermions and bosons and the two distributions for gases. MaxwellBoltzmann gases, velocity distribution and fermiDirac gases are investigated as the module provides an uncomplicated and direct approach to the subject, using frequent illustrations from low temperature physics.
Students will provide a unified survey of the statistical physics of gases, including a full treatment of quantum statistics, gaining a fuller insight into the meaning of entropy. Students will gain knowledge in applications of statistics to various types of gas. Ultimately, students will develop the ability to apply expressions and distributions in order to form accurate deductions, for example using the Boltzmann distribution for the probability of finding a system in a particular quantum state. Additionally, students will learn the role of statistical concepts in understanding macroscopic systems, and will be able to describe superfluidity in liquid helium, BoseEinstein condensation and black body radiation.

Theoretical Physics Group Project
In the Theoretical Physics Group Project, students will work as part of a team and will receive guidance on project management, planning and meetings. The module will culminate in a writingup stage in which the groups will prepare a group report, and students will be presented an opportunity to give an individual talk at the physics miniconference.
The module equips students with the ability to develop a theoretical physics research project with formulation, literature searches, data gathering, analysis and presentation.

Theoretical Physics with Mathematics Independent Study
This module requires independent study in various aspects of theoretical physics is guided by a series of workshops. Students will solve problems, partake in intense reading, and prepare and deliver presentations. Students will be provided with an opportunity to extend their preliminary studies by undertaking openended investigations into various aspects and problems of theoretical physics, writing up their findings in an individual report.
The module will teach analytical recipes of theoretical physics used in quantum mechanics, with the focus on the variational functions method, operator techniques with applications in perturbation theory methods and coherent states of a quantum harmonic oscillator, whilst students will train in the use of the operator algebra of 'creation' and 'annihilation' operators in the harmonic oscillator problem, which will develop a basis for the introduction of second quantisation in manybody systems. Students will develop an understanding in the algebra of creation and annihilation operators for Bose and Fermi systems, along with exploring the secondquantised representation of Hamiltonians of interacting manybody systems.
Additionally, students will analyse BoseEinstein condensation in one, two, and threedimensional systems and will develop the ability to describe the condensate using the method of coherent states, as well as covering the GinzburgLandau theory of a superfluid phase transition and to describe vortices in a superfluid. They will learn to relate Bose and Fermi statistics to the symmetry of manybody systems with respect to permutations of identical particles and will receive computer programming skills, as well as gaining experience in individual project work. The module’s main aim is to prepare students to enable them to undertake fourth year theoretical physics projects.
Core

Differential Equations
Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.
While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.

Elliptic Curves
Students will be given a solid foundation in the basicsof algebraic geometry. They will explore how curves can be described by algebraic equations, and learn how to use abstract groups in dealing with geometrical objects (curves). The module will present applications and results of the theory of elliptic curves and provide a useful link between concepts from algebra and geometry.
Students will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally, they will learn to perform algebraic computations with elliptic curves.

Geometry of Curves and Surfaces
The topic of smooth curves and surfaces in threedimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.
A number of wellknown concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

Groups and Symmetry
Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.
Situations will be considered in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.
Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.

Hilbert Space
Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.
Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.

Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.

Linear Systems
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a (A,B,C,D) system.

Metric Spaces
An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.
They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

Number Theory
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the nonspecialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Representation Theory of Finite Groups
Students will cover the basics of ordinary representation theory. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between the two will be discussed.
The second part is an introduction to the ordinary character theory of finite groups, intrinsic to representation theory. Students will learn the concepts of Rmodule and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups.
They will also learn to perform computations with representations and morphisms in a selection of finite groups

Rings Fields and Polynomials
Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.
They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
How wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.
Optional
Year 4

MPhys Literature Review
Information for this module is currently unavailable.

MPhys Project
MPhys projects vary from year to year and are tailored to suit the individual student and the available research facilities. This twomodule project commences with a dissertation or literature review. Students will write a report on the project work and will conduct a presentation for the miniconference in the summer term, along with gaining skills related to oral presentation of scientific research.
Project work gives students the opportunity to carry out research or a detailed investigation into a specific area of physics appropriate to their chosen degree theme. Students will develop and apply analytical and problemsolving skills in an open ended situation, involving use of the library, computer, and other resources as appropriate, working alone or in a small group.
By the end of this module, students will demonstrate the ability to plan, manage and execute an investigation an area of physics in a systematic way using appropriate techniques. They will formulate conclusions and critically compare with relevant theory, and may be required to generate and analyse data and critically assess experimental uncertainties.
Core

Advanced Electrodynamics and Gravity
This module introduces a range of geometry and exterior calculus, including scalar fields, vector fields and convector fields. Students will explore pforms, exterior derivative, metrics and Hodge dual, and will discover electrodynamics, more specifically Maxwell equations in terms of the Maxwell 2form, 4velocity fields and Lorentz force equation in terms of the Maxwell 2form. Gravity is also covered, and students will engage in topics such as Einstein 3forms, stressenergymomentum 3forms and Einstein equations. Additionally, students will gain knowledge of killing vectors, spacetimes with symmetry, conserved quantities and black holes.
Students will gain the knowledge required to display an understanding of the intrinsic, covariant nature of electrodynamic, along with a familiarity with handling the Einstein equations and field equations on curved spacetime. Students will also be able to formulate and tackle field theories on spacetime using tools from modern differential geometry.

Advanced Magnetism
Students will be offered a revision of elements of the theory of electromagnetism, before being introduced to the phenomenology of solid state magnetic phenomena. The module discusses Van Vleck's description of diamagnetism and diamagnetism as quantum phenomenon. Students will explore ferromagnetism and antiferromagnetism, ferromagnetic exchange and the Heisenberg model, which includes selfconsistent mean field theory. A description of ferromagnetic phase transitions and Curie temperature will be provided as part of the module, along with the elements of the GinzburgLandau theory of magnetic phase transitions.
By the end of the module, students will develop a knowledge and understanding of magnetic and electric phenomena in condensed matter physics, in addition to an enhanced awareness of recent advances and current problems in condensed matter physics.

Advanced Relativity and Gravity
The module offers a short review of special relativity, tensor calculus on Minkowski spacetime, differential calculus on Minkowski spacetime, and curved spacetimes. Students will explore general relativity, gravity as intrinsic curvature of spacetime, and the Einstein equations, along with predictions of the linearized Einstein equations, gravitational waves, and gravitomagnetic field equations. Students will investigate exact solutions of the Einstein equations, black holes and event horizons.
By the end of the module, students will have a basis knowledge and understanding of the theories of special and general relativity, and possess a conceptual understanding of the links between Newtonian mechanics and relativity. The module also provides a geometrical insight into the properties of spacetime and relativity.

Elliptic Curves
This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.
Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.

Galois Theory
Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a ‘smallest possible’ field L containing k and the roots of f; and a finite group G which describes certain ‘allowed’ permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1to1 correspondence with the subgroups of G.
In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.

Hilbert Space
Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.
The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.

Lasers and Applications
This module will address the necessary requirements for laser action, spontaneous and stimulated emission rates, Einstein coefficients, optical gain coefficient, and characteristics of the emitted light. Students will become aware of the different types of lasers, such as gas and solid state, semiconductor, dye, chemical and excimer lasers. Semiconductor lasers: homojunction, single and double heterojunction devices will be investigated, along with materials and operating requirements. The module explores fabrication methods, quantum well lasers, advantages and characteristics. There will be a focus on a range of applications including laser surgery, optical fibre communications, laser machining, pollution monitoring and remote sensing.
By the end of the module, students will be familiarised with lasers and their applications, including the operating principles of a variety of different lasers. Students will understand the many uses of lasers in industry, medicine and the environment.

Lebesgue Integration
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.

Lie Groups and Lie Algebras
The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.
The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.

Matter at low temperature
The module begins by discussing what physicists mean by high and low temperatures, and looks at the different types of ordering that may occur as systems cool. Students will explore cryogenic techniques used for accessing such low temperatures are described, including the design of useful cryostats. Students will observe the new phenomena that occur when systems are cooled below room temperature and will consider electron pairing leading to the zero resistance of superconducting materials, the effect of magnetic fields, and the role of macroscopic quantum mechanical wave functions. The module provides an overview of the practical uses in superconducting quantum interference devices (SQUIDs).
The module seeks to explore a selection of fascinating phenomena that occurs when cooling matter to temperatures more than a million times colder than the familiar 290K of everyday life and observe the significance for both physics and technology. Additionally, students will appreciate the relation between temperature and order, will know how low temperatures are produced, including dilution refrigerators, and will also be able to describe the phenomena of superconductivity and superfluidity.

Operator Theory
Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems. Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.
First, students will review Hilbert spaces, before spending some time studying infinitedimensional operators, notably the unilateral shift and multiplication operators, aswell as basic concepts. They will then consider the criteria for invertibilityof selfadjoint operators, leading to the spectral theory of such operators.

Probability Theory
The aim of this module is to develop an analytical and axiomatic approach to the theory of probabilities.
Students will consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. They will then use random variables and the expectation to develop a probability calculus, which is applied to achieve laws of large numbers for sums of independent random variables. Finally the characteristic function will be used to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.

Quantum transport in Low Dimensional Nanostructures
Students can expect to explore twodimensional electron systems, quantum transport in disordered lowdimensional electron systems and semiconductor quantum wires. The LandauerButtiker conductance formula is investigated, focusing on impurities in quantum wires, electronic transport in a magnetic field and the Hall effect. The module considers metallic point contacts, the pointcontact spectroscopy of the electronphonon interaction, and atomic breakjunctions and the scanning tunnelling microscope. Students will receive examples of applications of scanning tunnelling microscopy as part of the module.
By the end of the module, students will have knowledge of the physics of nanoscale solid state devices and how these may be manufactured and utilised. They will enhance their awareness of the recent advancements and current problems in condensed matter physics.

Representation Theory of Finite Groups
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of Rmodule and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.
The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.
The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.

Semiconductor Device Physics
Students will familiarise themselves with crystal growth, including growth theory, faceting, impurity segregation and zone refining. The module presents students with a silicon case study, investigating semiconducting properties, silicon oxide, masking, surface pacification and photolithographic processing. Compound semiconductors will be discussed, covering band structure advantages over silicon, IIVI materials and effects of iconicity.
Additionally, students will explore thin film semiconductors, such as epitaxy, vapour phase growth, metalloorganic methods and liquid phase epitaxy, and the module provides a broad interdisciplinary overview of the linkage between the physics, chemistry and other materials sciences involved in the synthesis of semiconductors and the devices made from them.
By the end of the module, students will develop an understanding of the basic properties of crystals and crystal defects, and will be able to describe how crystals are grown and discuss the main semiconductor used for microelectronics as a detailed case study. Students will also demonstrate how physics continues to play a major role in enabling information technology.

Topology and Fractals
Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.
This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you’ll consider topological ideas of open and closed sets in the setting of R^2.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.

Calculus

Careers
Careers
Physics is an exciting subject that is fundamental to the developments in modern society. Applications of the subject range from the very pure to the very practical, and a physics degree opens up a wide range of rewarding careers in scientific research and technological development, as well as in a variety of other professions. A substantial number of our graduates continue on to postgraduate education, or enter employment that directly relies on their specialist skills. Our students also find employment in careers where they are valued because of general skills gained during the course such as logical thinking, problem solving, numeracy and computer literacy. Examples include consulting, finance, computer programming, and accountancy, as well as managerial and administrative positions.
Lancaster University is dedicated to ensuring you not only gain a highly reputable degree, you also graduate with the relevant life and work based skills. We are unique in that every student is eligible to participate in The Lancaster Award which offers you the opportunity to complete key activities such as work experience, employability/career development, campus community and social development. Visit our Employability section for full details.

Fees and Funding
Fees and Funding
Fees
We set our fees on an annual basis and the 2019/20 entry fees have not yet been set.
As a guide, our fees in 2018 were:
UK/EU Overseas £9,250 £18,890 Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students taking Part I of the course are charged £10 for a bound copy of the lab manual. There are a number of optional oneday visits to places of interest and students pay travel costs.
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.
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10.1 hours
Typical time in lectures, seminars and similar per week during term time
42%
Average assessment by coursework
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