Physics and Mathematics BSc (Hons)

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This joint Physics and Mathematics honors program at the University of Liverpool equips students with a strong foundation in both disciplines, combining mathematical techniques with concepts such as quantum mechanics and relativity. Graduates are highly sought after, with excellent career prospects in industries like research, computing, and finance. The program emphasizes problem-solving skills and includes research-connected teaching, active learning, and authentic assessment, ensuring students develop into confident global citizens.

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Degree Overview:

Combining the study of Physics and Mathematics in your degree programme will give you a strong foundation for your future career. You will learn mathematical techniques to help you deal with new ideas and will understand new concepts such as quantium mechanics and relativity.

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Introduction

Mathematics is a fascinating, beautiful and diverse subject to study. It underpins a wide range of disciplines; from physical sciences to social science, from biology to business and finance. Physics is the most fundamental of the sciences. New concepts, such as quantum mechanics and relativity, are introduced at degree level in order to understand nature at the deepest level. These theories have profound philosophical implications because they challenge our view of the everyday world. At the same time they have a huge impact on society since they underpin the technological revolution. Combining the study of Physics and Mathematics in your degree programme will give you a strong mathematical training. You will learn mathematical techniques to help you to deal with new ideas that often seem counterintuitive, such as string theory, black holes, superconductors and chaos theory. Physics and Mathematics degrees are highly prized and our graduates have excellent career opportunities.

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What you'll learn

• Numeracy
• Problem solving skills
• Ability to reason and communicate clearly
• Teamwork
• Presentation skills

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Outline:

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Year One

In your first year you will take core mathematics modules, a module in Newtonian Mechanics, and physics modules. Many quantities can be expressed as the limiting value of a sequence of approximations, for example the slope of a tangent to a curve, the rate of change of a function, the area under a curve, and so on. Calculus provides us with tools for studying all of these, and more. Many of the ideas can be traced back to the ancient Greeks, but calculus as we now understand it was first developed in the 17th Century, independently by Newton and Leibniz. The modern form presented in this module was fully worked out in the late 19th Century. MATH101 lays the foundation for the use of calculus in more advanced modules on differential equations, differential geometry, theoretical physics, stochastic analysis, and many other topics. It begins from the very basics – the notions of real number, sequence, limit, real function, and continuity – and uses these to give a rigorous treatment of derivatives and integrals for real functions of one real variable.

• CALCULUS II (MATH102): This module, the last one of the core modules in Year 1, is built upon the knowledge you gain from MATH101 (Calculus I) in the first semester.
The syllabus is conceptually divided into three parts: Part I, relying on your knowledge of infinite series, presents a thorough study of power series (Taylor expansions, binomial theorem); part II begins with a discussion of functions of several variables and then establishes the idea of partial differentiation together with its various applications, including chain rule, total differential, directional derivative, tangent planes, extrema of functions and Taylor expansions; finally, part III is on double integrals and their applications, such as finding centres of mass of thin bodies. As is described in the module, this failure was first seen in interactions at the atomic scale and was first seen in experiments involving atoms and electrons. The module shows how Newton’s ideas were replaced by Quantum mechanics, which has been critical to explaining phenomena ranging from the photo-electric effect to the fluctuations in the energy of the Cosmic Microwave Background. The module also explains how this revolution in physicist’s thinking paved the way for developments such as the laser. No previous computing experience is assumed. During the course of the module, students are guided through a series of structured exercises which introduce them to the Python programming language and help them acquire a range of skills including: plotting data in a variety of ways; simple Monte Carlo techniques; algorithm development; and basic symbolic manipulations. The exercises are based around the content of the first year physics modules, both encouraging students to recognise the relevance of computing to their physics studies and enabling them to develop a deeper understanding of aspects of their first year course.
• Introduction to Linear Algebra (MATH103): Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces.
It is the study of lines, planes, and subspaces and their intersections using algebra. Linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Cramer’s Rule for solving linear systems was devised in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination. All these classical themes, in their modern interpretation, are included in the module, which culminates in a detailed study of eigenproblems. A part of the module is devoted to complex numbers which are basically just planar vectors. Linear algebra is central to both pure and applied mathematics. This module is an essential pre-requisite for nearly all modules taught in the Department of Mathematical Sciences. It introduces the basic principles like conservation of momentum and energy, and leads to the quantitative description of motions of bodies under simple force systems. It includes angular momentum, rigid body dynamics and moments of inertia.
• PRACTICAL SKILLS FOR MATHEMATICAL PHYSICS (PHYS156): This is a practical-based module exclusively for students taking joint maths and physics degree programmes.
In the sessions you will work through progressively more challenging experiments with increasingly complex equipment. You may work alone or in a pair, but you will be supported by a demonstrator who will give you a lot of feedback on your work. In the classes you will be expected to contribute to class discussions and put your results on a whiteboard. There is an assessment associated with each laboratory practical. Again, there will be a variety of activities that will allow you to demonstrate different parts of your learning. You will also have to write at least two full reports for which you will receive written and verbal feedback.
• Thermal Physics and Properties of Matter (PHYS102): Einstein said in 1949 that "Thermodynamics is the only physical theory of universal content which I am convinced, within the areas of applicability of its basic concepts, will never be overthrown."
In this module, different aspects of thermal physics are addressed: (i) classical thermodynamics which deals with macroscopic properties, such as pressure, volume and temperature – the underlying microscopic physics is not included; (ii) kinetic theory of gases describes the properties of gases in terms of probability distributions associated with the motions of individual molecules; and (iii) statistical mechanics which starts from a microscopic description and then employs statistical methods to derive macroscopic properties. The laws of thermodynamics are introduced and applied.
• Electricity, Magnetism and Waves (PHYS103): Electricity, Magnetism and Waves lie at the heart of physics, being phenomena associated with almost every branch of physics including quantum physics, nuclear physics, condensed matter physics and accelerator physics, as well as numerous applied aspects of physics such as communications science.
The course is roughly divided into two sections. The second part involves the study of oscillations and waves and focuses on solutions of the wave equation, the principles of superposition, and examples of wave phenomena.

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Year Two

All modules in year two are compulsory.

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Compulsory Modules:

• CLASSICAL MECHANICS (MATH228): This module is concerned with the motion of physical bodies both in everyday situations and in the solar system.
To describe motion, acceleration and forces you will need background knowledge of calculus, differentiation, integration and partial derivatives from MATH101 (Calculus I), MATH102 (Calculus II) and MATH103 (Introduction to Linear Algebra). Classical mechanics is important for learning about modern developments such as relativity (MATH326), quantum mechanics (MATH325) and chaos and dynamical systems (MATH322). This module will make you familiar with notions such as energy, force, momentum and angular momentum which lie at the foundations of applied mathematics problems.
• COMPLEX FUNCTIONS (MATH243): This module introduces students to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.
• Electromagnetism I (PHYS201): The study of classical electromagnetism, one of the fundamental physical theories.
Mathematical methods shall be developed and exercised for the study of physical systems.
• Quantum and Atomic Physics I (PHYS203): The course aims to introduce 2nd year students to the concepts and formalism of quantum mechanics.
The Schrodinger equation is used to describe the physics of quantum systems in bound states (infinite and finite well potentials, harmonic oscillator, hydrogen atoms, multi-electron atoms) or scattering (potential steps and barriers). Basis of atomic spectroscopy are also introduced.
• Differential Equations (MATH221): Differential equations play a central role in mathematical sciences because they allow us to describe a wide variety of real-world systems and the mathematical techniques encountered in this module are useful to a number of later modules; this is why MATH201 is compulsory for a number of degree programmes.
The module will aim to stress the importance of both theory and applications of ordinary differential equations (ODEs) and partial differential equations (PDEs), putting a strong emphasis on problem solving and examples. It has broadly 5 parts and each part contains two types of equations: those that can be solved by specific methods and others that cannot be solved but can only be studied to understand some properties of the underlying equations and their solutions. The main topics are first order ODEs, second order ODEs, systems of ODEs, first-order PDEs and some of the most well-known second-order PDEs, namely the wave, heat and Laplace equations. It is not the study of the very small (particle and nuclear physics) or the very large (astrophysics and cosmology) but of the things in between.
• Nuclear and Particle Physics (PHYS204): This module introduces the basic properties of particles and nuclei, their stability, modes of decay, reactions and conservation laws.
Recent research in particle physics is highlighted, and for nuclear physics some of the applications (such as nuclear power) are given. This module leads on to more specialist optional modules in Year 3, in particle physics, nuclear physics and nuclear power.

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Year Three

Exactly one of the modules MATH325 and PHYS361 must be taken. Exactly one of the project modules MATH334 and PHYS379 must be taken and passed.

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Compulsory Modules:

• FURTHER METHODS OF APPLIED MATHEMATICS (MATH323): Ordinary and partial differential equations (ODEs and PDEs) are crucial to many areas of science, engineering and finance.
This module addresses methods for, or related to, their solution. It starts with a section on inhomogeneous linear second-order ODEs which are often required for the solution of higher-level problems. We then generalize basic calculus by considering the optimization of functionals, e.g., integrals involving an unknown function and its derivatives, which leads to a wide variety of ODEs and PDEs. After those systems of two linear first-order PDEs and second-order PDES are classified and reduced to ODEs where possible. In certain cases, e.g., `elliptic’ PDEs like the Laplace equation, such a reduction is impossible. The last third of the module is devoted to two approaches, conformal mappings and Fourier transforms, which can be used to obtain solutions of the Laplace equation and other irreducible PDEs.
• Relativity (MATH326): Einstein’s theories of special and general relativity have introduced a new concept of space and time, which underlies modern particle physics, astrophysics and cosmology.
It makes use of, and has stimulated the development of modern differential geometry. This module develops the required mathematics (tensors, differential geometry) together with applications of the theory to particle physics, black holes and cosmology. The module places emphasis on the importance of conservation laws in integral form, and on the fundamental role integral conservation laws play in the derivation of partial differential equations used to model different physical phenomena in problems of solid and fluid mechanics.
• QUANTUM MECHANICS (MATH325): The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time.
• More Is Different: Statistical Mechanics, Thermodynamics, and All That (MATH327): Statistical Physics is a core subject in Physics and a cornerstone for modern technologies.
To name just one example, quantum statistics is informing leading edge developments around ultra-cold gases and liquids giving rise to new materials. After successfully completing this module students will understand statistical ensembles and related concepts such as entropy and temperature, will understand the properties of classical and quantum gases, will be know the laws of thermodynamics and will be aware of advanced phenomena such as phase transition. The module will also develop numerical computer programming skills for the description of macroscopic effects such as diffusion by an underlying stochastic process.
• Numerical Methods for Ordinary and Partial Differential Equations (MATH336): Many real-world systems in mathematics, physics and engineering can be described by differential equations.
In rare cases these can be solved exactly by purely analytical methods, but much more often we can only solve the equations numerically, by reducing the problem to an iterative scheme that requires hundreds of steps. We will learn efficient methods for solving ODEs and PDEs on a computer. In complex dynamics, we consider the case where the state of the system is described by a single (complex) variable, and the rule of evolution is given by a holomorphic function. In the course of this study, we will encounter many results about complex functions that may seem “magic” when compared with what might be expected from real analysis. A highlight of this kind is the theorem that every polynomial is “chaotic” on its Julia set.
• DIFFERENTIAL GEOMETRY (MATH349): Differential geometry studies distances and curvatures on manifolds through differentiation and integration.
This module introduces the methods of differential geometry on the concrete examples of curves and surfaces in 3-dimensional Euclidean space. The module MATH248 (Geometry of curves) develops methods of differential geometry on examples of plane curves. This material will be discussed in the first weeks of the course, but previous familiarity with these methods is helpful. Students following a pathway in theoretical physics might find this module interesting as it discusses a different aspect of differential geometry, and might take it together with MATH326 (Relativity). MATH410 (Manifolds, homology and Morse theory) and MATH446 (Lie groups and Lie algebras).
• MATHEMATICAL PHYSICS PROJECT (MATH334): This is one of the project choices for students in Year 3 of MMath Mathematical Physics (FGH1) and MPhys Theoretical Physics (F344) degree programmes, the other choice being the Physics project module PHYS305.
Students perform research in an interesting topic in Mathematical Physics under the supervision of a member of staff, which is followed by preparation of a report and an oral presentation. This project will provide insights into more advanced subjects and experience in handling specialist literature.
• PRACTICAL PHYSICS III (PHYS306): Year 3 Laboratory.
• PHYSICS INTERNSHIP (PHYS309): The physics internship module is designed to give students the experience of working in a STEM related working environment or setting that is different from any project work that they undertake in the Department of Physics.
It should provide an insight into how students may apply skills and experiences later in their career; whether working abroad or in any other non-UoL, off-campus scientific or secondary school setting. It deals with the study of the structure and physical properties of large collection of atoms that compose materials, which are found in nature or synthesized in laboratory. This particular module aims to advance and extend the concepts on solids introduced in Year 1 and Year 2 modules. Especially, it focuses on the atomic structure and behaviour of electrons in crystalline materials, which are essential for understanding of physical phenomena in complex systems. Collective models of the nucleus leading to a quantitative understanding of rotational and vibrational excitations are developed. Finally, electromagnetic decays between excited states are introduced as spectroscopic tools to probe and understand nuclear structure.
• Materials Physics and Characterisation (PHYS387): Preparation and characterisation of a range of materials of scientific and technological importance.
• Physics Data Analysis with Statistics (PHYS392): Statistical Methods in Physics Analysis: Understanding Statistics and its application to data analysis
• Statistical Thermodynamics (PHYS393): The problem to understand blackbody radiation opened the door to modern physics.
A statistical understanding of thermodynamic quantities will be developed together with a method of deriving thermodynamic potentials from the properties of the quantum system. Applications are shown in solid state physics and the Planck blackbody radiation spectrum. The description of simple systems will be covered before extending to real systems. Perturbation theory will be used to determine the detailed physical effects seen in atomic systems.
• Accelerators and Radioisotopes in Medicine (PHYS246): This module provides an introduction to applications of accelerators and radioisotopes in medical imaging and tumour therapy.
The lectures are complemented by workshops in which students can work collaboratively on problems to solve set problems. Experimental demonstrations to reinforce concepts also take place in the workshops. As well as being of interest to students considering careers in medical physics or nuclear-related industries, this module should also appeal to those curious to see how physics can be applied in a multidisciplinary approach to other areas of science.
• Computational Modelling (PHYS305): Computational methods are at the heart of many modern physics experiments and mastering these techniques is invaluable also beyond fundamental research.
In this module we introduce students to object-oriented concepts of a modern programming language (Python) and employ this to model experiments. A combination of Monte Carlo methods (based on random trials) and deterministic methods to solve differential equations are used. Students will then apply their knowledge in a small-group project connected to the state-of-the-art research done in the department. The project topics are taken from different areas of particle, nuclear or accelerator physics and range from analyses situated at the Large Hadron Collider to medical applications of proton beams.
• Electromagnetism II (PHYS370): The module builds on first and second year modules on electricity, magnetism and waves to show how a wide variety of physical phenomena can be explained in terms of the properties of electromagnetic radiation.
The main part of the course is focussed on cosmology, which is study of the content of the universe, structure on the largest scales, and its dynamical evolution. This is covered from both a theoretical and observational perspective.
• Particle Physics (PHYS377): Introduction to Particle Physics.
To build on the second year module involving Nuclear and Particle Physics.
• Surfaces and Interfaces (PHYS381): This module gives a brief introduction into the physics of solid surfaces their experimental study.
Surfaces and interfaces are everywhere and many surface-related phenomena are common in daily life (texture, friction, surface-tension, corrosion, heterogeneous catalysis). What happens to the electronic properties and vibrational properties upon creating a surface? What happens in detail when we adsorb an atom or a molecule on a surface? After reviewing the underlying physics principles, the design and operation and nuclear fission reactors is introduced. The possibility of energy from nuclear fusion is then discussed, with the present status and outlook given. The project work is typically based in one of the research groups within the Department of Physics. There is also the option to pursue individual projects in cooperation with an industrial partner. The student will be planning, managing and accomplishing an extended investigation of a physics-based or physics-related problem under the supervision of one or more academic staff members. In case of an industry-based project, there are two supervisors required, one academic and one from industry. BSc projects may be experimental, observational, computational, theoretical or educational. The output of the project will be written up in a project report and presented in the form of a talk. Industry-based projects can be related to any in-house developments but not to an actual product release. The programme is to develop graduates to acquire skills in: development of solving new complex tasks; initiative and creativity; communication and cooperation with others; project organisation and self management. Quantitative scientific skills will be emphasized so as to make graduates of the course gain a wider experience of report writing displaying high standards of composition and production. This builds on prior knowledge of thermodynamics, fluid behaviour and semiconductors to show how these concepts can be practically applied to power generation and storage systems.
• Magnetic Properties of Solids (PHYS399): The magnetic properties of solids are exploited extensively in a wide range of technologies, from hard disk drives, to sensors, to magnetic resonance imaging, and the development of magnetic materials is a multi-billion pound industry.
Fundamentally, magnetism in condensed matter also represents one of the best examples of quantum mechanics in action, even at room temperature and on a macroscopically observable scale. In this module we will explore how the interactions between electrons in solids can result in the magnetic moment, and how this relates to the quantum mechanical property of spin.
• Mathematical Biology (MATH335): In the current age of big data, mathematics is becoming indispensable in order for us to make sense of experimental results and in order to gain a deeper understanding into mechanisms of complex biological systems.
This module will focus on teaching students how to construct and analyse models for a wide range of biological systems. Mathematical approaches covered will be widely applicable.
• Mathematics of Networks and Epidemics (MATH338): Networks are familiar to us from many real-world systems such as the internet, power grids, transportation and biological networks.
The underpinning mathematical concept is called a graph an it is no surprise that the same issues arise in each area, whether this is to identify the most important or influential individuals in the network, or to prevent dynamics on the network (e.g. epidemics) or to make the network robust to the dynamics it supports (e.g. power grids and transportation). In this module, we learn about different classes of networks and how to quantify and describe them including their structures and their nodes. We will consider real-world biological applications of network theory, in particular focusing on epidemics.

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Assessment:

Most modules are assessed by a two and a half hour examination in January or May, but many have an element of coursework assessment. This might be through homework, class tests, mini-project work or key skills exercises.

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Teaching:

Your learning activities will consist of lectures, tutorials, practical classes, problem classes, private study and supervised project work. In year one, lectures are supplemented by a thorough system of group tutorials and computing work is carried out in supervised practical classes. Key study skills, presentation skills and group work start in first-year tutorials and are developed later in the programme. The emphasis in most modules is on the development of problem solving skills, which are regarded very highly by employers. Project supervision is on a one-to-one basis, apart from group projects in year two.

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Careers:

Physics and Mathematics degrees are highly prized and our graduates have excellent career opportunities in industrial research and development, computing, business, finance and teaching.

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Typical types of work our graduates have gone onto include:

• An actuarial trainee analyst
• A graduate management trainee risk analyst
• A trainee chartered accountant

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Recent employers of our graduates are:

• Barclays Bank plc
• Deloitte
• Forrest Recruitment
• Marks and Spencer
• Mercer Human Resource Consulting Ltd

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Other:

We have a large department with highly qualified staff, a first-class reputation in teaching and research, and a great city in which to live and work. Your course will be delivered by the Department of Mathematical Sciences.

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Supporting your learning

• Careers and employability support, including help with career planning, understanding the job market and strengthening your networking skills
• Confidential counselling and support to help students with personal problems affecting their studies and general wellbeing

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Liverpool Hallmarks

We have a distinctive approach to education, the Liverpool Curriculum Framework, which focuses on research-connected teaching, active learning, and authentic assessment to ensure our students graduate as digitally fluent and confident global citizens.

• Digital fluency
• Confidence
• Global citizenship
• Research-connected teaching
• Active learning
• Authentic assessment
All this is underpinned by our core value of inclusivity and commitment to providing a curriculum that is accessible to all students.

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UK Fees:

Full-time place, per year £9,250 Year in industry fee £1,850 Year abroad fee £1,385

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International Fees:

Full-time place, per year £27,200 Year abroad fee £13,600 Fees shown are for the academic year 2024/25. Please note that the Year Abroad fee also applies to the Year in China. Tuition fees cover the cost of your teaching and assessment, operating facilities such as libraries, IT equipment, and access to academic and personal support.