Embark on a virtual, 360-degree, Department tour and get a sneak peek of our dedicated labs and study areas!
Mathematics at Lancaster
Discover what studying Mathematics at Lancaster is like from our students and academics.
5th for Physics and Astronomy
The Times and Sunday Times Good University Guide (2023)
5th for Physics
The Guardian University Guide (2023)
10th for Physics
The Complete University Guide (2023)
Taught jointly with Lancaster’s Department of Mathematics and Statistics, our BSc 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 your final year you also carry out an investigative group project, and have a choice of options such as Quantum Information and Advanced Gravity and Relativity.
Theoretical Physics and Theoretical Physics with Mathematics are disciplines that are fundamental to advancements in modern society. Theoretical Physicists are highly numerate with advanced problem-solving skills, programming knowledge, critical thinking abilities and project management experience, all of which are developed and refined over the course of your degree. These skills open a wealth of career options from the very pure, such as expanding knowledge through scientific research, or very practical like exploring the world of data science and software development. Many of our graduates continue their studies to PhD level and embark on a career in academia. A wealth of additional opportunities also exists, such as teaching or careers within the business and finance sectors. Our graduates are well-paid, with the median starting salary of our Physics degrees being £25,500 (HESA Graduate Outcomes Survey 2022).
Here are just some of the roles that our BSc and MPhys Theoretical Physics and Theoretical Physics with Mathematics students have progressed into upon graduating:
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.
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.
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 firstname.lastname@example.org
Lancaster University offers a range of programmes, some of which follow a structured study programme, and some which offer the chance for you to devise a more flexible programme to complement your main specialism.
Information contained on the website with respect to modules is correct at the time of publication, and the University will make every reasonable effort to offer modules as advertised. In some cases changes may be necessary and may result in some combinations being unavailable, for example as a result of student feedback, timetabling, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes and new research. Not all optional modules are available every year.
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.
In Classical Mechanics students will apply the fundamental ideas of Newtonian mechanics to important systems such as rotating bodies, planetary systems and classical fluids. The focus is on gravitation and its central importance in determining the large-scale behaviour of the Universe. Concepts such as inertial and gravitational mass, black holes and dark matter will be explored.
This module will also consider how to extend the principles of basic kinematics and dynamics to rotational situations, giving students an understanding of torque, moment of inertia, centre of mass, angular momentum and equilibrium. The fundamental mechanical principles will then be applied to understand basic properties of materials including elasticity of solids and fluid dynamics.
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.
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.
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, Ampere's law and Faraday's law of electromagnetic induction.
This course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.
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 and temperature, and the direction of the wind. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.
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.
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 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.
The ultimate description of the Universe requires quantum and not classical mechanics. This module begins 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. They will consider 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.
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.
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.
In this module students will 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.
The Special Theory of Relativity will be introduced, beginning with Einstein's postulates and moving on to inertial reference frames, the physics of simultaneity, length contraction, time dilation, relativistic energy and momentum, and space-time diagrams.
This module focuses on the study of the thermal properties of matter, during which 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, 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.
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 three-dimensional 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.
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 way to 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.
This module provides students with a working knowledge and understanding of electromagnetism 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. The module addresses the basic properties of wave propagation, diffraction and inference, and laser operation.
Students will develop their skills in vector calculus and will learn to apply Maxwell’s equations in the analysis of common electromagnetic phenomena, including motors and generators, and in the fundamental connection between electricity and magnetism. Students will gain practical knowledge of Fresnel and Fraunhofer diffraction, as well as thin-film interference fringes and anti-reflection 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.
This module expands students' knowledge of topics such as Newton's laws and the solution of dynamical problems. It introduces the Lagrangian formulation of mechanics and explains its relation to Newton's equations and the least-action principle. The 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 complement the taught material. Students 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 applications to problems in physics and beyond, and exploit the generality of the 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.
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 structure-preserving 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.
This module covers quantum mechanics in all its generality, from its central postulates and mathematical language to concrete phenomena and applications. Students will learn how the main postulates give precise meaning to the states, observables, dynamics, and measurable properties of quantum systems, and how this translates into general features of the theory, such as its inherent probabilistic nature on the one hand, and precise quantisation of observable properties on the other. This material is introduced and developed by a series of model systems and applications, such as the particle in a box, tunneling through a barrier, the harmonic oscillator, the hydrogen atom, angular momentum and spin, driven systems displaying radiative transitions, and many-body systems exhibiting entanglement and obeying the Pauli exclusion principle. Students will also become familiar with the mathematical language of the theory, including the Dirac notation, differential and matrix operators, commutation relations, and the role of eigenvalue problems. They will also gain practice with solution techniques such as separation of variables, stationary and time-dependent perturbation theory, the Ehrenfest and Heisenberg picture. Through further applications and examples, students will acquire prerequisite knowledge for advanced courses in atomic and molecular physics, condensed matter physics, particle physics, astrophysics, and quantum information processing.
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 A-level.
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 A-level, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.
Students will receive an introductory concepts-based approach to the physics of special relativity, nuclei and fundamental particles. The module covers the general properties of nuclei, such as composition, the forces within the nucleus, mass, binding energy and nuclear decay. Students are then introduced to the standard model of particle physics, including the three generations of fundamental particles and their interactions.
Students will gain an understanding of the basis of Einstein’s theory of special relativity in electro-magnetism, both conceptually and mathematically, and 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.
This module introduces one-electron atoms and the spin-orbit 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 non-conservation.
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.
The module offers an introduction to electronic, thermal, optical and magnetic properties of solids.
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, crystal lattices and phonons, reciprocal lattice and diffraction of waves, the electronic band structure in metals, insulators and semiconductors. Students will explore electronic properties of semiconductors, including the physics of p-n junctions and their optical properties. Students will be introduced to the basics of magnetism in solids.
Students will gain an enhanced understanding of solid state physics, and will be able to describe major pieces of experimental evidence supporting the key theoretical ideas, including the experimental techniques used.
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. Maxwell-Boltzmann gases, velocity distribution and fermi-Dirac 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, Bose-Einstein condensation and black body radiation.
The Theoretical Physics Group project is linked to all of our Theory degrees.
In the Theoretical Physics Group Project, students will work as part of a team to undertake an open-ended investigation of a Theoretical Physics-based problem. The project is not tightly-restrained by defined limits, and students have to make decisions about the direction of their research. Recent project areas include machine learning, quantum computing, chaos, and the use of cellular automata to model the spread of disease or forest fires. Students will receive guidance on project management, planning and meetings, and the module will culminate in the submission of a group written report and an individual talk at the physics student conference, The PLACE. The module equips students with the ability to develop a theoretical physics research project with literature searches, formulation, modelling, analysis and presentation.
This module requires students to undertake an independent study in various aspects of theoretical physics. It provides an opportunity for students to extend their preliminary studies by undertaking open-ended investigations into various aspects/problems of theoretical physics. Students will write up their findings in a report.
This module aims to 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. Students will be trained 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 many-body systems. In addition, the module will introduce the algebra of creation and annihilation operators for Bose and Fermi systems, along with second-quantised representation of Hamiltonians of interacting many-body systems. Students will learn to apply a mathematical basis of complex analysis in order to solve problems in mathematical and theoretical physics.
Students will be given a solid foundation in the basics of 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.
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 well-known 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.
Building on the skills developed in the Scientific Programming and Modelling project, this module will introduce students to new elements of Java, and will involve more sophisticated modelling of physical systems, such as calculating the range of a cannon ball, and simulating the motion of the moon around the earth.
Students will develop a more thorough knowledge of the java language, including the use of inheritance, and will be able to write a physics modelling program in java.
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.
The module introduces students to energy demand in the past, present and future, looking at energy use by sector and country. Students will study thermal power stations, nuclear power and take a planetary view of energy sources. From there, the module moves to renewable energy, costing energy and looking at Hydrogen as a fuel for the future. Students will consider energy use in the home and at work, looking at energy efficiency and alternative small-scale energy sources.
By the end of the module, students will gain a broad overview of energy and the issues involved from a physical basis, and will be able to clearly explain the physics of energy and global warming and make an informed contribution to the debate.
The topic of smooth curves and surfaces in three-dimensional 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 well-known 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.
The study of graphs - mathematical objects used to model pairwise relations between objects - is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.
The module is compulsory on the Particle Physics, Particle Physics with Cosmology and Theoretical Physics Pathways and an option on all others.
The module will cover various topics, such as symmetries and transformations; groups, group invariants and generators. As well as this, students will learn about irreducible representations; orthogonal groups O(2) and O(3); unitary groups SU(2) and SU(3) and applications to spin, isospin, colour and flavour of elementary particles.
By the end of the module, students will have a basic knowledge and understanding of the concepts and methods used in group theory. They will be able to apply these concepts and methods to problems in particle physics, cosmology and field theory.
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.
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.
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.
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.
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.
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.
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.
Introducing continuum mechanics, this module focuses on body and contact force, global balance laws, and decomposition of the contact force into shear and pressure components.
Students will explore static fluids, ideal fluids and the Euler equation. The module then examines Newtonian fluids, waves and the two-fluid model of plasmas.
Students will be introduced to fluid dynamics and its applications within physics, and will develop an understanding of the origin, solution and application of Navier-Stokes equations, along with the wider applications of the Navier-Stokes theory to bio-, geo- and astrophysical systems. Students will also solve problems based on the application of the general principles of the physics of fluids.
This module focuses on what constitutes life. It explores the stability and synchronisation in complex and open interacting systems, entropy and information, and DNA as an information storage system. Students will investigate fundamental rate processes, ion channel mechanics and molecular diffusion and Brownian motion. In addition, cellular structure and function, along with membrane potential and action potential are studied, and the module examines the functioning of the cardiovascular system as an information-processing system and the interactions between cardiovascular oscillations and brain waves.
Students will develop an awareness of how physical principles help to understand the function of living systems at various levels of complexity, as well as an appreciation that living systems are structures in time as much as structures in space.
Ultimately, the module will equip students with the ability to explain the basic characteristics of living systems as thermodynamically open systems, in addition to teaching the physical principles of the functioning of a cell, how cells make ensembles (tissues and organs), and how they interact within larger biological systems. Students will then apply their knowledge of physics and mathematics to the understanding of basic principles of living systems – starting from a cell to the cardiovascular system and the brain.
The module is compulsory on the Particle Physics and Particle Physics with Cosmology Pathways and an option on all others.
This module covers various topics, including the CKM matrix and its parameterisations, unitarily constraints and the unitarity triangle and the status of experimental measurements, theory and observations of neutrino oscillations. Students will also study CP violation and current heavy flavour particle physics topics, such as c- and b-hadron production and decay analysis, along with top quark physics.
Students will develop a basic knowledge of the phenomenology of flavour mixing in the quark sector and neutrino oscillations, and they will gain an awareness of the concepts of transformation, invariance and symmetry and their mathematical descriptions. Additionally, students will reinforce their understanding of the basic ideas, concepts and analyses of the experimental data on flavour mixing in weak interactions of hadrons and neutrino oscillations, in addition to gaining knowledge of some current topics on the physics of heavy flavours, which are likely directions of the experimental particle physics research in Lancaster.
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.
The module consolidates the theoretical concepts of quantum information processing, exploring Dirac notation, density matrices and evolution, and entanglement. Students will also explore qubits, quantum algorithms, circuit design and error connection. In addition, the module will address trapped ions and atoms, Josephson junctions and quantum optics.
By the end of the module, students will be familiar with the fundamental concepts of quantum processing, such as density matrices and the dynamics of quantum systems, and will be able to understand how these can be implemented in realistic devices. Students will learn about experimental implementation based on atom-optical realisations and realisations in the solid state, and will apply these to explore theoretical concepts that have a vast area of application in condensed matter physics and atom-quantum-optics.
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 R-module 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
This module is compulsory in the Quantum Pathway, and is an option on the other pathways.
This module will provide students with an overview of solid-state realisations of quantum technologies, including superconductivity, low-dimensional structures, and impurity and donor systems, and introduces students to fabrication and characterisation techniques for micro-structures and nano-structures.
Students will develop skills to formulate problems in precise terms, while identifying key issues, to solve problems and provide well-defined solutions. Students will also understand how to collate and understand complex information from a range of sources, including verbal information from lectures, lecture notes, and key textbooks.
The module is compulsory on the Astrophysics with Space Pathway and an option on all other Pathways.
Space and Auroral Physics explores the physics of the solar-terrestrial environment, from the solar wind, which streams from the Sun towards Earth, to how the atmosphere couples to the local space environment. This module will introduce you to basic plasma physics, the dynamics of Earth’s magnetosphere, and the formation of the aurora. It will also address the causes and impacts of space weather of technology and society.
We set our fees on an annual basis and the 2024/25 entry fees have not yet been set.
As a guide, our fees in 2023/24 were:
There are a number of optional one-day visits to places of interest and students pay travel costs.
There may be extra costs related to your course for items such as books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation, you may need to pay a subscription to a professional body for some chosen careers.
Specific additional costs for studying at Lancaster are listed below.
Lancaster is proud to be one of only a handful of UK universities to have a collegiate system. Every student belongs to a college, and all students pay a small college membership fee which supports the running of college events and activities.
For students starting in 2022 and 2023, the fee is £40 for undergraduates and research students and £15 for students on one-year courses. Fees for students starting in 2024 have not yet been set.
To support your studies, you will also require access to a computer, along with reliable internet access. You will be able to access a range of software and services from a Windows, Mac, Chromebook or Linux device. For certain degree programmes, you may need a specific device, or we may provide you with a laptop and appropriate software - details of which will be available on relevant programme pages. A dedicated IT support helpdesk is available in the event of any problems.
The University provides limited financial support to assist students who do not have the required IT equipment or broadband support in place.
In addition to travel and accommodation costs, while you are studying abroad, you will need to have a passport and, depending on the country, there may be other costs such as travel documents (e.g. VISA or work permit) and any tests and vaccines that are required at the time of travel. Some countries may require proof of funds.
In addition to possible commuting costs during your placement, you may need to buy clothing that is suitable for your workplace and you may have accommodation costs. Depending on the employer and your job, you may have other costs such as copies of personal documents required by your employer for example.
Details of our scholarships and bursaries for 2024-entry study are not yet available, but you can use our opportunities for 2023-entry applicants as guidance.
Check our current list of scholarships and bursaries.
Join Meenal and Vlad as they take you on a tour of the Lancaster University campus. Discover the learning facilities, accommodation, sports facilities, welfare, cafes, bars, parkland and more.Undergraduate Open Days
The information on this site relates primarily to 2024/2025 entry to the University and every effort has been taken to ensure the information is correct at the time of publication.
The University will use all reasonable effort to deliver the courses as described, but the University reserves the right to make changes to advertised courses. In exceptional circumstances that are beyond the University’s reasonable control (Force Majeure Events), we may need to amend the programmes and provision advertised. In this event, the University will take reasonable steps to minimise the disruption to your studies. If a course is withdrawn or if there are any fundamental changes to your course, we will give you reasonable notice and you will be entitled to request that you are considered for an alternative course or withdraw your application. You are advised to revisit our website for up-to-date course information before you submit your application.
More information on limits to the University’s liability can be found in our legal information.
We believe in the importance of a strong and productive partnership between our students and staff. In order to ensure your time at Lancaster is a positive experience we have worked with the Students’ Union to articulate this relationship and the standards to which the University and its students aspire. View our Charter and other policies.