# Mathematical Physics MMath

- Course length: 4 years
- UCAS code: FGH1
- Year of entry: 2022
- A-level requirements: AAB

## Honours Select

×This programme offers Honours Select combinations.

## Honours Select 100

×This programme is available through Honours Select as a Single Honours (100%).

## Honours Select 75

×This programme is available through Honours Select as a Major (75%).

## Honours Select 50

×This programme is available through Honours Select as a Joint Honours (50%).

## Honours Select 25

×This programme is available through Honours Select as a Minor (25%).

## Study abroad

×This programme offers study abroad opportunities.

## Year in China

×This programme offers the opportunity to spend a Year in China.

## Accredited

×This programme is accredited.

### Module details

**Due to the impact of COVID-19 we're changing how the course is delivered.**

### Programme Year One

** **

#### Year One Compulsory Modules

##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.3. To introduce the notions of sequences and series and of their convergence.

**Learning Outcomes**(LO1) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

(LO2) Apply the methods of calculus and real analysis to solve previously unseen problems (of a similar style to those covered in the course).

(LO3) Understand in interpret proofs in the context of real analysis and apply the theorems developed in the course to straightforward examples.

(LO4) Independently construct proofs of previously unseen mathematical results in real analysis (of a similar style to those demonstrated in the course).

(LO5) Differentiate and integrate a wide range of functions;

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

(S1) Numeracy

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**To discuss local behaviour of functions using Taylor’s theorem.

To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals.**Learning Outcomes**(LO1) Use Taylor series to obtain local approximations to functions

(LO2) Obtain partial derivatives and use them in several applications such as, error analysis, stationary points change of variables.

(LO3) Evaluate double integrals using Cartesian and Polar Co-ordinates.

##### Foundations of Quantum Physics (PHYS104)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To introduce the concepts and the experimental foundations of quantum theory. To carry out simple calculations related to quantum mechanical problem tasks. To show the impact of quantum theory on contemporary science.

**Learning Outcomes**(LO1) An understanding why classical mechanics must have failed to describe the properties of light, and the properties of microspopic systems.

(LO2) An understanding of why quantum theory is the conceptual framework required to explain the behaviour of the universe.

(LO3) A basic knowledge on the experimental and theoretical concepts which founded modern physics, i.e. quantum theory needed to explain certain phenomena.

(LO4) An understanding of the quantum theory of light and the ability to apply energy-momentum conservation in the explanation of phenomena such as the. photo-electric effect and the Compton effect.

(LO5) An understanding of de Broglie waves and their interpretation.

(LO6) An ability to explain the experimental evidence for de Broglie waves, for example through the scattering of electrons, X-rays and neutrons.

(LO7) An understanding of the principles of quantum mechanical measurements and Heisenberg's uncertainty principle.

(LO8) An understanding of the identity principle of microscopic particles and the basic idea of quantum (Fermi-Dirac and Bose-Einstein) statistics.

(LO9) An understanding why quantum theory is the conceptual framework to understand the microscopic properties of the universe.

(LO10) A basic knowledge of contemporary applications of quantum theory and their impact on our society.

(LO11) A basic understanding of the Schrodinger equation.

(LO12) An understanding of de Broglie waves and their statistical interpretation.

(LO13) An understanding of Bohr's theory of the atom and its application to the H-atom including the concept of principal quantum numbers.

(S1) Problem solving skills relating to quantum phenomena.

##### Introduction to Computational Physics (PHYS105)

**Level**1 **Credit level**7.5 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**To develop the ability to break down physical problems into steps amenable to solution using algorithms To develop skills in using computers to perform and run algorithms To introduce techniques for analysing and presenting data To introduce elemenatry Monte Carlo techniques To introduce basic computer algebra To illustrate the insight into physics which can be obtained using computational methods

**Learning Outcomes**(LO1) Ability to produce algorithms to solve simple physical problems.

(LO2) Ability to program and use simple algorithms on a computer

(LO3) Ability to analyse and present physical data

(LO4) Ability to produce simple Monte Carlo models

(LO5) Ability to carry out basic symbolic manipulations using a computer

(S1) Problem solving skills

(S2) Communication skills

(S3) IT skills

##### Introduction to Linear Algebra (MATH103)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**• To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.

• To develop geometrical intuition in 2 and 3 dimensions.

• To introduce students to the concept of subspace in a concrete situation.

• To provide a foundation for the study of linear problems both within mathematics and in other subjects**Learning Outcomes**(LO1) Manipulate complex numbers and solve simple equations involving them, solve arbitrary systems of linear equations.

(LO2) Understand and use matrix arithmetic, including the computation of matrix inverses.

(LO3) Compute and use determinants.

(LO4) Understand and use vector methods in the geometry of 2 and 3 dimensions.

(LO5) Calculate eigenvalues and eigenvectors.

(S1) Numeracy

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems. Learning Outcomes: After completing the module students should be able to analyse real world problems involving: - the motions of bodies under simple force systems - conservation laws for momentum and energy - rigid body dynamics using centre of mass, angular momentum and moments of inertia

**Learning Outcomes**(LO1) the motions of bodies under simple force systems

(LO2) conservation laws for momentum and energy

(LO3) rigid body dynamics using centre of mass, angular momentum and moments

(LO4) oscillation, vibration, resonance

(S1) Representing physical problems in a mathematical way

(S2) Problem Solving Skills

##### Practical Skills for Mathematical Physics (PHYS156)

**Level**1 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**To develop skills with spreadsheets; to develop skills in using computers to perform mathematical calculations; to illustrate the insight into physics which can be obtained by exploiting computational software packages; to improve science students' skills in communicating scientific information in appropriate written and oral formats; to provide a core of essential introductory laboratory methods which overlap and develop from A-level; to introduce the basis of experimental techniques in physical measurement, the use of computer techniques in analysis and to provide experience doing experiments, keeping records and writing reports.

**Learning Outcomes**(LO1) Ability to use spreadsheets and mathematical packages to calculate and graph mathematical equations.

(LO2) Ability to apply mathematical software packages to physics problems.

(LO3) Appreciation of how to present results by computer.

(LO4) Ability to communicate more confidently.

(LO5) Understanding of some of the key factors in successful communication.

(LO6) Appreciation of the practical nature of physics.

(LO7) Awareness of the importance of accurate experimentation, particularly obervation and record keeping.

(LO8) Ability to plan, execute and report on the results of an investigation using appropriate analysis of the data and associated uncertainties.

(LO9) Practical and technical skill required for physics experimentation and an appreciation of the importance of a systematic approach to experimental measurement.

(S1) Problem solving skills.

(S2) Communication skills.

(S3) IT skills.

(S4) Analytical Skills - ability to grasp complex concepts, to understand and interpret data precisely and to construct logical arguments. Ability to distil a problem to its basic elements.

##### Thermal Physics and Properties of Matter (PHYS102)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**The module aims to make the student familiar with

• The concepts of Thermal Physics

• The zeroth, first and second laws of Thermodynamics

• Heat engines

• The kinetic theory of gases

• Entropy

• The equation of state

• Van der Waals equation

• States of matter and state changes

• Mechanical properties of solids

• The basis of statistical mechanics**Learning Outcomes**(LO1) Be able to link the microscopic view of a system to its macroscopic state variables

(LO2) Be able to derive and use Maxwell's relations

(LO3) Calculate the linear and volume thermal expansions of materials

(LO4) Analyse the expected performance of heat engines, heat pumps and refrigerators

(LO5) Calculate the heat flow into and work done by a system and how that is constrained by the first law of thermodynamics

(LO7) Understand the PV and PT diagrams for materials and the phase transitions that occur when changing the state variables for materials

(LO9) Use the theory of equipartition to relate the structure of molecules to the measured heat capacity

(LO10) Relate the second law of thermodynamics to the operation of heat engines, heat pumps and refrigerators, particularly the Carnot engine

(LO11) Understand the kinetic theory of gases and calculate properties of gases including the heat capacity and mean free path

(LO12) Understand the basis of entropy and relate this to the second law of thermodynamics and calculate entropy changes

(S1) Problem Solving Skills

##### Wave Phenomena (PHYS103)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To introduce the fundamental concepts and principles of wave phenomena. To highlight the many diverse areas of physics in which an understanding of waves is crucial. To introduce the concepts of interference and diffraction.

**Learning Outcomes**(LO1) At the end of the module, the should be able to:Demonstrate an understanding of oscillators.

(LO2) Understand the fundamental principles underlying wave phenomena.

(LO3) Apply those principles to diverse phenomena.

(LO4) Understand wave reflection and transmission, superposition of waves.

(LO5) Solve problems on the behaviour of electromagnetic waves in vacuo and in dielectric materials.

(LO6) Understand linear and circular polarisation.

(LO7) Understand inteference and diffraction effects.

(LO8) Understand lenses and optical instruments.

(LO9) Apply Fourier techniques and understand their link to diffraction patterns.

(LO10) Understand the basic principles of lasers

(S1) Problem solving

### Programme Year Two

** **

#### Year Two Compulsory Modules

##### Classical Mechanics (MATH228)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To provide an understanding of the principles of Classical Mechanics and their application to dynamical systems.

**Learning Outcomes**(LO1) To understand the variational principles, Lagrangian mechanics, Hamiltonian mechanics.

(LO2) To be able to use Newtonian gravity and Kepler's laws to perform the calculations of the orbits of satellites, comets and planetary motions.

(LO3) To understand the motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravity over the Earth's surface.

(LO4) To understand the connection between symmetry and conservation laws.

(LO5) To be able to work with inertial and non-inertial frames.

(S1) Applying mathematics to physical problems

(S2) Problem solving skills

##### Complex Functions (MATH243)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To introduce the student 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.

**Learning Outcomes**(LO1) To understand the central role of complex numbers in mathematics;.

(LO2) To develop the knowledge and understanding of all the classical holomorphic functions.

(LO3) To be able to compute Taylor and Laurent series of standard holomorphic functions.

(LO4) To understand various Cauchy formulae and theorems and their applications.

(LO5) To be able to reduce a real definite integral to a contour integral.

(LO6) To be competent at computing contour integrals.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### Condensed Matter Physics (PHYS202)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**The aims of this module are to introduce the most important and basic concepts in condensed matter physics relating to the different materials we commonly see in the world around us. Condensed matter physics is one of the most active areas of research in modern physics, whose scope is extremely broad. The ultimate aim of this module is to introduce its central ideas and methodology to the students.

**Learning Outcomes**(LO1) Students will have the knowledge and skills to understand the basic concepts of bonding in solids, establish an understanding of electron configuration in atoms and in the condensed matter in terms of bonding, and relating them to band structure description.

(LO2) Students will be able to understand how solid structures are described mathematically and how material properties can be predicted.

(LO3) Students will be able to establish a foundation in basic crystallography, using Bragg's law, and understand the concept of the reciprocal lattice.

(LO4) Students will understand basic transport properties, both electronic and thermal, in solids.

(LO5) Students will understand the concept of electron and hole carrier statistics, effective masses and transport in intrinsic and extrinsic semiconductors

(LO6) Students will learn the basics of magnetism, the atomic origin and classical treatment of diamagnetism and paramagnetism, quantization of angular momentum and Hund's rule, and introduced to weak magnetism in solids.

(LO7) Students will become familiar to the general language of condensed matter physics, key theories and concepts, ultimately enebling them to read and understand research papers.

##### Electromagnetism I (PHYS201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To introduce the fundamental concepts and principles of electrostatics, magnetostatics, electromagnetism, Maxwell's equations, and electromagnetic waves; to introduce differential vector analysis in the context of electromagnetism; to introduce circuit principles and analysis (EMF, Ohm's law, Kirchhoff's rules, RC and RLC circuits); to introduce the formulation fo Maxwell's equations in the presence of dielectric and magnetic materials; to develop the ability of students to apply Maxwell's equations to simple problems involving dielectric and magnetic materials; to develop the concepts of field theories in Physics using electromagnetism as an example; to introduce light as an electromagnetic wave.

**Learning Outcomes**(LO1) Demonstrate a good knowledge of the laws of electromagnetism and an understanding of the practical meaning of Maxwell's equations in integral and differential forms.

(LO2) Apply differential vector analysis to electromagnetism.

(LO3) Demonstrate simple knowledge and understanding of how the presence of matter affects electrostatics and magnetostatics, and the ability to solve simple problems in these situations.

(LO4) Demonstrate knowledge and understanding of how the laws are altered in the case of non-static electric and magnetic fields and the ability to solve simple problems in these situations.

(S1) Problem solving skills.

(S2) Analytic skills applied to the study of electromagnetic phenomena.

(S3) Mathematical skills applied for the development of deep intuition on electromagnetic phenomena and to the study of physical systems.

##### Nuclear and Particle Physics (PHYS204)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To introduce Rutherford and related scattering; to introduce nuclear size, mass and decay modes; to provide some applications and examples of nuclear physics; to introduce particle physics, including interactions, reactions and decay; to show some recent experimental discoveries; to introduce relativistic 4-vectors for applications to collision problems.

**Learning Outcomes**(LO1) A basic understanding of Rutherford, electron on neutron scattering.

(LO2) An understanding of the basic principles that determine nuclear size, mass and decay modes.

(LO3) The knowledge of examples and applications of nuclear physics.

(LO4) An understanding of the basic properties of particles and their interactions

(LO5) An understanding of conservation laws and their role in particle decays and reactions

(LO6) A basic understanding of relativistic 4-vectors

(LO7) A basic understanding of drawing Feynman diagrams. Knowledge of some particle physics results: neutrino physics, measurement of top quark and W masses, structure of the proton

(LO8) Knowledge of particle physics results: Large hadron collider, cosmic microwave background, dark matter, super-symmetry

##### Quantum and Atomic Physics I (PHYS203)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To introduce students to the concepts of quantum theory. To show how Schrodinger's equation is applied to bound states (well potentials, harmonic oscillator, hydrogen atoms, multi-electron atoms) and particle flux (scattering) . To show how quantum ideas provide an understanding of atomic structure.

**Learning Outcomes**(LO1) At the end of the module the student should have an understanding of the reasons why microscopic systems require quantum description and statistical interpretation.

(LO2) At the end of the module the student should have knowledge of the Schrodinger equation and how it is formulated to describe simple physical systems.

(LO3) At the end of the module the student should have understanding of the basic technique of using Schrodinger's equation and ability to determine solutions in simple cases.

(LO4) At the end of the module the student should have understanding of how orbital angular momentum is described in quantum mechanics and why there is a need for spin.

(LO5) At the end of the module the student should have understanding how the formalism of quantum mechanics describes the structure of atomic hydrogen and, schematically, how more complex atoms are described.

##### Vector Calculus With Applications in Fluid Mechanics (MATH225)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them. To give an appreciation of the many applications of vector calculus to physical situations. To provide an introduction to the subjects of fluid mechanics and electromagnetism.

**Learning Outcomes**(LO1) After completing the module students should be able to: - Work confidently with different coordinate systems. - Evaluate line, surface and volume integrals. - Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes. - Recognise the many physical situations that involve the use of vector calculus. - Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow. All learning outcomes are assessed by both examination and course work.

##### Differential Equations (MATH221)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**•To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.

•To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.

**Learning Outcomes**(LO1) To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.

(LO2) To know the elementary techniques for the solution of ODEs.

(LO3) To understand the idea of reducing a complex ODE to a simpler one.

(LO4) To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.

(LO5) To understand a range of applications of ODE.

(S1) Problem solving skills

(S2) Numeracy

#### Year Three Compulsory Modules

##### Further Methods of Applied Mathematics (MATH323)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**•To give an insight into some specific methods for solving important types of ordinary differential equations.

•To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

•To build on the students'' existing knowledge of partial differential equations of first and second order.

**Learning Outcomes**(LO1) After completing the module students should be able to:

- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.- solve simple integral extremal problems including cases with constraints;

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.

[This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]##### Relativity (MATH326)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**(i) To introduce the physical principles behind Special and General Relativity and their main consequences;

(ii) To develop the competence in the mathematical framework of the subjects - Lorentz transformation and Minkowski space-time, semi-Riemannian geometry and curved space-time, symmetries and conservation laws, Variational principles.

(iii) To develop the understanding of the dynamics of particles and of the Maxwell field in Minkowski space-time, and of particles in curved space-time

(iv) To develop the knowledge of tests of General Relativity, including the classical tests (perihelion shift, gravitational deflection of light)

(v) To understand the basic concepts of black holes and (time permitting) relativistic cosmology and gravitational waves.

**Learning Outcomes**(LO1) To be proficient at calculations involving Lorentz transformations, the kinematical and dynamical quantities associated to particles in Minkowski space-times, and the application of the conservation law for the four-momentum to scattering processes.

(LO2) To know the relativistically covariant form of the Maxwell equations .

(LO3) To know the action principles for relativistic particles, the Maxwell field and the gravitational field.

(LO4) To be proficient at calculations in semi-Riemannian geometry as far as needed for General Relativity, including calculations involving general coordinate transformations, tensor fields, covariant derivatives, parallel transport, geodesics and curvature.

(LO5) To understand the arguments leading to the Einstein's field equations and how Newton's law of gravity arises as a limiting

(LO6) To be able to calculate the trajectories of bodies in a Schwarzschild space-time.

(S1) problem solving skills

(S2) numeracy

#### Year Three Optional Modules

##### Quantum Mechanics (MATH325)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**The aim of the module is to lead the student to an understanding of the way that relatively simple mathematics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

**Learning Outcomes**(LO1) To be able to solve Schrodinger's equation for simple systems.

(LO2) To have an understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

(S1) Problem solving skills

(S2) Numeracy

##### Quantum and Atomic Physics II (PHYS361)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To build on the second year module on Quantum and Atomic Physics.

To develop the formalism of quantum mechanics.

To develop an understanding that atoms are quantum systems.

To enable the student to follow elementary quantum mechanical arguments in the literature**Learning Outcomes**(LO1) Understanding of the role of wavefunctions, operators, eigenvalue equations, symmetries, compatibility/non-compatibility of observables and perturbation theory in quantum mechanical theory.

(LO2) An ability to solve straightforward problems - different bound states and perturbing interactions.

(LO3) Developed knowledge and understanding of the quantum mechanical description of atoms - single particle levels, coupled angular momentum, fine structure, transition selection rules.

(LO4) Developed a working knowledge of interactions, electron configurations and coupling in atoms.

(S1) Problem solving skills

(S2) Analytic skills applied to quantum systems

##### Mathematical Physics Project (MATH334)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**To study in depth an area of theoretical physics and report on it.

**Learning Outcomes**(LO1) After completing the project with suitable guidance the student should have

· understood an area of advanced theoretical physics

· had experience in consulting relevant literature

· gained experience in using appropriate mathematics

· made a critical appraisal of the current understanding of the area · learnt how to construct a written essay and given an oral presentation##### Computational Modelling (PHYS305)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**• To revise Python programming skills and reinforce object-oriented concepts and methods of a high-level Object-oriented programming language.

• To apply Python for the computational modelling of physical phenomena and solution of complex physics problems using Monte Carlo techniques and numerical integration.

• To further develop the ability to efficiently implement algorithms using Python and verify the results.

• To give students experience of working independently and in small groups on an original problem.

• To give students an opportunity to display the high quality of their work, initiative and ingenuity.

• To give students experience of report writing displaying high standards of composition and production.

• To give an opportunity for students to further develop and display oral communication skills.

**Learning Outcomes**(LO1) Acquire a deep knowledge of a high level programming language including object-oriented elements.

(LO2) Gain experience how to apply computational methods to the solution of physics problems, including the set up of a complex model of physical phenomena or experimental situation

(LO3) Experience in researching literature and other sources of relevant information

(LO4) Experience in testing model against data from experiment or literature

(LO5) Improved ability to organise and manage time.

(LO6) Improved skills in report writing.

(LO7) Improved skills in explaining project under questioning.

(S1) Problem solving skills

(S2) Teamwork

(S3) Organisational skills

(S4) Communication skills

(S5) IT skills

##### Physics Internship (PHYS309)

**Level**3 **Credit level**15 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**Provide students with an insight into the process of scientific research and debate or communicating science in a STEM-related setting different from the University of Liverpool;

Expose students to new research, cultural and working environments;

Develop the confidence to work independently and in a team, to effectively and efficiently apply science to attain a STEM-related goal;

Develop students’ ability to communicate scientific concepts and findings in a variety of formats;

Develop students' employability skills.

**Learning Outcomes**(LO1) to maintain accurate records of experiments or classroom related experiences, and reliable and comprehensive account of any methodologies used

(LO2) to prepare and deliver oral presentations to high scientific and professional standards that describes the experiences during the internship, the research objectives and the rationale behind the project design.

(LO3) to write a professional report on the project priorities, the internal and external drivers of the project strategy and the potential impact of the project on the local and wider community.

(LO4) to analyse and evaluate data, information and experiences and to draw valid conclusions while working in a professional environment.

(LO5) to identify and articulate their personal and professional transferable skills and connect them to their employability.

##### Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.

**Learning Outcomes**(LO1) To understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations.

(LO2) To apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### Solid State Physics (PHYS363)

**Level**3 **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To develop concepts introduced in Year 1 and Year 2 modules which relate to solids; to consolidate concepts related to crystal structure; to introduce the concept of reciprocal space and diffraction; to enable the students to apply these concepts to the description of crystals,transport properties and the electronic structure of condensed matter; to illustrate the use of these concepts in scientific research in condensed matter; to introduce various other solids.

**Learning Outcomes**(LO1) Familiarity with the crystalline nature of both perfect and real materials.

(LO2) An understanding of the fundamental principles of the properties of condensed matter.

(LO3) An appreciation of the relationship between the real space and the reciprocal space view of the properties of crystalline matter.

(LO4) An ability to describe the crystal structure and electronic structure of matter

(LO5) An awareness of current physics research in condensed matter.

(S1) An ability to describe the crystal structure and electronic structure of matter.

##### Nuclear Physics (PHYS375)

**Level**3 **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To build on the second year module involving Nuclear Physics; to develop an understanding of the modern view of nuclei, how they are modelled and of nuclear decay processes.

**Learning Outcomes**(LO1) Knowledge of evidence for the shell model of nuclei, its development and the successes and failures of the model in explaining nuclear properties.

(LO2) Knowledge of the collective vibrational and rotational models of nuclei.

(LO3) Basic knowledge of nuclear decay processes, alpha decay and fission, of gamma-ray transitions and internal conversion.

(LO4) Knowledge of electromagnetic transitions in nuclei.

(S1) How to use mathematics to describe the physical world.

(S2) How to tackle problems in physics and formulate an appropriate solution.

(S3) How to compare results critically with predictions from theory.

##### Practical Physics III (PHYS306)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To give further training in laboratory techniques, in the use of computer packages for modelling and analysis, and in the use of modern instruments. To develop independent judgement in performing physics experiments. To encourage students to research aspects of physics complementary to material met in lectures and tutorials. To consolidate the students ability to produce good quality work against realistic deadlines.

**Learning Outcomes**(LO1) Experience of taking physics data with modern equipment

(LO2) Knowledge of experimental techniques not met in previous laboratory practice

(LO3) Improved skills in researching published papers and articles as source materials

(LO4) Developed a personal responsibility for assuring that data taken are of a high quality

(LO5) Increased skills in data taking and error analysis

(LO6) Increased skills in reporting experiments and an appreciation of the factors needed to produce clear and complete reports

(LO7) Improved skills in the time management and organisation of their experimental procedures to meet deadlines

(S1) Problem solving skills

(S2) Organisational skills

##### Materials Physics and Characterisation (PHYS387)

**Level**3 **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**• To teach the properties and methods of preparation of a range of materials of scientific and technological importance

• To develop an understanding of the experimental techniques of materials characterisation

• To introduce materials such as amorphous solids, liquid crystals and polymers and to develop an understanding of the relationship between structure and physical properties for such materials

• To illustrate the concepts and principles by reference to examples**Learning Outcomes**(LO1) An understanding of the atomic structure in crystalline and amorphous materials

(LO2) Knowledge of the methods used for preparing single crystals and amorphous materials

(LO3) Knowledge of the experimental techniques used in materials characterisation

(LO4) Knowledge of the physical properties of superconducting, liquid crystal and polymer materials

(LO5) An appreciation of the factors involved in the design of biomaterials

(S1) Problem solving skills

##### Semiconductor Applications (PHYS389)

**Level**3 **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To develop the physics concepts describing semiconductors in sufficient details for the purpose of understanding the construction and operation of common semiconductor devices

**Learning Outcomes**(LO1) At the end of the module the student should have: Knowledge of the basic theory of p-n junctions Knowledge of the structure and function of a variety of semiconductor devices An overview of semiconductor device manufacturing processes Knowledge of the basic processes involved in the interaction of radiation with matter Understanding the application of semiconductors in Nuclear and Particle physics

##### Statistics for Physics Analysis (PHYS392)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To give a theoretical and practical understanding of the statistical principles involved in the analysis and interpretation of data. To give practice in analysing data by computer program. To show how to write code to solve problems in data analysis.

**Learning Outcomes**(LO1) Knowledge of experimental errors and probability distributions

(LO2) The ability to use statistical methods in data analysis

(LO3) The ability to apply statistical analysis to data from a range of sources

(LO4) Using statistical information to determine the validity of a hypothesis or experimental measurement

(LO5) The ability to write code to analyse data sets

(S1) Problem solving skills

(S2) Numeracy

(S3) Digital scholarship participating in emerging academic, professional and research practices that depend on digital systems

(S4) IT skills

##### Statistical Physics (PHYS393)

**Level**3 **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**• To build on material presented in earlier Thermal Physics and Quantum Mechanics courses

• To develop the statistical treatment of quantum systems

• To use theoretical techniques to predict experimental observables

• To introduce the basic principles governing the behaviour of liquid helium and superconductors in cooling techniques**Learning Outcomes**(LO1) Understanding of the statistical basis of entropy and temperature

(LO2) Ability to devise expressions for observables, (heat capacity, magnetisation) from statistical treatment of quantum systems

(LO3) Understanding of Maxwell Boltzmann, Fermi-Dirac and Bose Einstein gases

##### Statistical Physics (MATH327)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. To develop an understanding of the foundations of Statistical Physics normally including statistical ensembles and related extensive and intrinsic quantities.

2. To develop an understanding of the properties of classical and quantum gases and an appreciation of their applications to concepts such as the classical equation of state or the statistical

theory of photons.

3. To obtain a reasonable level of skill in using computer simulations for describing diffusion and transport in terms of stochastic processes.

4. To know the laws of thermodynamics and thermodynamical cycles.

5. To obtain a reasonable understanding of interacting statistical systems and related phenomenons such as phase transitions.**Learning Outcomes**(LO1) Demonstrate understanding of the microcanonical, canonical and grand canonical ensembles, their relation and the derived concepts of entropy, temperature and particle number

density.(LO2) Understand the derivation of the equation-of-state for non-interacting classical or quantum gases.

(LO3) Demonstrate numerical skills to understand diffusion from an underlying stochastic process.

(LO4) Know the laws of thermodynamics and demonstrate their application to thermodynamic cycles.

(LO5) Be aware of the effect of interactions including an understanding of the origin of phase transitions.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

(S4) Communication skills

(S5) IT skills

(S6) Organisational skills

(S7) Teamwork

##### Game Theory (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur. To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc. To treat fully a number of specific games including the famous examples of "The Prisoners' Dilemma" and "The Battle of the Sexes". To treat in detail two-person zero-sum and non-zero-sum games. To give a brief review of n-person games. In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved.To see how the Prisoner's Dilemma arises in the context of public goods.

**Learning Outcomes**(LO1) To extend the appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

(LO2) To be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

(LO3) To be able to solve mathematically a variety of standard problems in the theory of games and to understand the relevance of such solutions in real situations.

##### Numerical Methods for Ordinary and Partial Differential Equations (MATH336)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**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.

**Learning Outcomes**(LO1) Demonstrate an advanced knowledge of the analysis of ODEs and PDEs underpinning the scientific programming within our context.

(LO2) Demonstrate an extended understanding of scientific programming and its application to numerical analysis and to other branches of Mathematics.

(LO3) Continuous engagement with putting practical problems into mathematical language.

(S1) Numeracy

(S2) Problem solving skills

(S3) Programming skills

##### The Magic of Complex Numbers: Complex Dynamics, Chaos and the Mandelbrot Set (MATH345)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects;

2. To introduce students to some topics of current and recent research in the field;

3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting;

4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a real-variable context, and the implications of this in complex dynamics;

5. To explain how complex-variable methods have been instrumental in questions purely about real-valued one-dimensional dynamical systems, such as the logistic family.

6. To deepen students' appreciations for formal reasoning and proof. After completing the module, students should be able to:

1. understand the compactification of the complex plane to the Riemann sphere, and use spherical distances and derivatives.

2. use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

3. state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

4. determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

5. apply advanced results from complex analysis in the setting of complex dynamics.

6. determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.**Learning Outcomes**(LO1) To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

(LO2) To be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

(LO3) To be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

(LO4) To be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

(LO5) To know how to apply advanced results from complex analysis in a dynamical setting.

(LO6) To be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

(S2) Problem solving skills

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space. While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

**Learning Outcomes**(LO1) 1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces.

(LO2) 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

(LO3) 1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.

(LO4) 1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

(LO5) 2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.

(LO6) 2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.

(LO7) 3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.

(LO8) 3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.

(LO9) 4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,

(LO10) 4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.

(S1) Problem solving skills

(S2) Numeracy

##### Introduction to String Theory (MATH423)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of string theory, and its utilization as a theory that unifies all of the known fundamental matter and interactions.

**Learning Outcomes**(LO1) After completing the module the students should: - be familiar with the properties of the classical string.

(LO2) be familiar with the basic structure of modern particle physics and how it may arise from string theory.

(LO3) be familiar with the basic properties of first quantized string and the implications for space-time dimensions.

(LO4) be familiar with string toroidal compactifications and T-duality.

(S1) Problem solving skills

(S2) Numeracy

##### Electromagnetism II (PHYS370)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To build on first and second year modules on electricity, magnetism and waves by understanding a range of electromagnetic phenomena in terms of Maxwell's equations; to understand the properties of solutions to the wave equation for electromagnetic fields in free space, in matter (non-dispersive and dispersive dielectrics, and conductors); to understand the behaviour of electromagnetic waves at boundaries; to understand the behaviour of electromagnetic waves in cavities, waveguides and transmission lines; to understand the properties of electric dipole radiation; to introduce an explicity covariant formulation of electromagnetism in special relativity'o further develop students' problem-solving and analytic skills.

**Learning Outcomes**(LO1) An understanding of the properties of solutions to the wave equation for electromagnetic fields in free space and in matter (non-dispersive and dispersive dielectrics, and conductors).

(LO2) An understanding of the behaviour of electromagnetic waves at boundaries.

(LO3) An understanding of the behaviour of electromagnetic waves in cavities, waveguides and transmission lines.

(LO4) An understanding of the properties of electric dipole radiation.

(LO5) The ability to explain an explicity covariant formulation of electromagnetism in special relativity.

(S1) Problem solving skills.

(S2) Numeracy.

##### Relativity and Cosmology (PHYS374)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To introduce the ideas of general relativity and demonstrate its relevance to modern astrophysics; to provide students with a full and rounded introduction to modern observational cosmology; to develop the basic theoretical background required to understand and appreciate the significance of recent results from facilities such as the Hubble Space Telescope and the Wilkinson Microwave Anisotropy Probe.

**Learning Outcomes**(LO1) The ability to explain the relationship between Newtonian gravity and Einstein's General Relativity (GR).

(LO2) Understanding of the concept of curved space time and knowledge of metrics.

(LO3) A broad and up-to-date knowledge of the basic ideas, most important discoveries and outstanding problems in modern cosmology.

(LO4) Knowledge of how simple cosmological models of the universe are constructed.

(LO5) The ability to calculate physical parameters and make observational predictions for a range of such models.

##### Particle Physics (PHYS377)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To build on the second year module Nuclear and Particle Physics

To develop an understanding of the modern view of particles, of their interactions and the Standard Model**Learning Outcomes**(LO1) At the end of the module the student should have: Basic understanding of relativistic kinematics (as applied to collisions, decay processes and cross sections)

(LO2) Descriptive knowledge of the Standard Model using a non rigorous Feynman diagram approach

(LO3) Knowledge of the fundamental particles of the Standard Model and the experimental evidence for the Standard Model

(LO4) Knowledge of conservation laws and discrete symmetries

(S1) Problem solving skills

(S2) Numeracy

##### Surfaces and Interfaces (PHYS381)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To develop a syllabus to describe the properties of surfaces; to convey an understanding of the physical properties of surfaces; to provide knowledge of a raneg of surface characterisation techniques; to illustrate surface processes and their relevance to technologies.

**Learning Outcomes**(LO1) To explain how the presence of the surface alters physical properties such as atomic an electronic structure.

(LO2) To choose the right characterisation technique to assess different surface properties.

(LO3) To have gained an appreciation of surface processes and their relevance to the modification of surface properties.

(LO4) To be able to describe surface alterations and processes using the right terminology.

(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

(S2) Problem solving skills.

##### Nuclear Power (PHYS376)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To develop an ability which allows educated and well informed opinions to be formed by the next generation of physicists on a wide range of issues in the context of the future energy needs of man. To describe and understand methods of utilising renewable energy sources such as hydropower, tidal power, wave power, wind power and solar power. To give knowledge and understanding of the design and operation of nuclear reactors To give knowledge and understanding of nuclear fusion as a source of power To give knowledge and understanding relevant to overall safety in the nuclear power industry To describe the origin of environmental radioactivity and understand the effects of radiation on humans

**Learning Outcomes**(LO1) Learned the fundamental physical principles underlying nuclear fission and fusion reactors

(LO2) Studied the applications of these principles in the design issues power generation

(LO3) An appreciation of the role of mathematics in modelling power generation

(LO4) Learned the fundamental physical principles concerning the origin and consequences of environmental radioactivity

(LO5) Developed an awareness of the safety issues involved in exposure to radiation

(LO6) Developed problem solving skills based on the material presented

(LO7) Developed an appreciation of the problems of supplying the required future energy needs and the scope and issues associated with the different possible methods

##### Energy Generation and Storage (PHYS372)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**The module aims to enable students to understand physical concepts related to key sources of energy generation and to carry out related analysis.

**Learning Outcomes**(LO1) Learned the fundamental physical principles underlying energy production using conventional and renewable energy sources

(LO2) Studied the applications of these principles in the design issues power generation

(LO3) An appreciation of the role of mathematics in modelling power generation

(LO4) Developed problem solving skills based on the material presented

(LO5) Developed an appreciation of the problems of supplying the required future energy needs and the scope and issues associated with the different possible methods

##### Magnetic Properties of Solids (PHYS399)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**Students will develop an understanding of the phenomena and fundamental mechanisms of magnetism in condensed matter. They will be able to assess and compare the quantum mechanical interactions at play in different solids, and their impact on observable magnetic properties.

**Learning Outcomes**(LO1) Atomic structure basis for magnetic moments.

(LO2) Definition of Magnetisation, magnetic susceptibility, diamagnetism, paramagnetism

(LO3) Magnetic moments of ions.

(LO4) Crystal fields and local environments

(LO5) Magnetic ordering, M vs T curve.

(LO6) Types of magnetic order: Ferromagnetism, antiferromagnetism, ferrimagnetism.

(LO7) Quantum origin of magnetism

### Programme Year Four

There is a large set of modules available, some of which are taught in alternate years. MMath/MPhys students will take at least seven of these during Years three and four. There is also a compulsory project.

#### Year Four Compulsory Modules

##### Advanced Quantum Physics (PHYS480)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To build on Y3 module on Quantum Mechanics and Atomic Physics (PHYS203) with the intention of providing breadth and depth in the understanding of the commonly used aspects of Quantum mechanics. To develop an understanding of the ideas of perturbation theory for complex quantum systems and of Fermi's Golden Rule. To develop an understanding of the techniques used to describe the scattering of particles. To demonstrate creation and annihilation operators using the harmonic oscillator as an example. To develop skills which enable numerical calculation of real physical quantum problem. To encourage enquiry into the philosophy of quantum theory including its explanation of classical mechanics.

**Learning Outcomes**(LO1) At the end of the module the student should have: Understanding of advanced quantum mechanical calculations (operators in matrix form, Dirac notation, etc.). Understanding of perturbation techniques. Understanding of transition and other matrix elements. Understanding of phase space factors. Understanding of partial wave techniques. Understanding of basic cross section calculations

(LO2) Understanding of examples of state-of-the art quantum physics experiments.

(LO3) Understanding of the implications of quantum physics in our daily lives

(S1) Problem solving skills

(S2) Analytic skills applied to quantum systems.

(S3) Communicating advanced physics problems.

##### Mathematical Physics Project (MATH420)

**Level**M **Credit level**30 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**To investigate and report on a topic at the boundary of current knowledge in theoretical physics.

**Learning Outcomes**(LO1) After completing the essay with suitable guidance, the student should have

· understood an area of current research in theoretical physics

· had experience in locating and consulting relevant research material, particularly through use of journals and the Internet

· learnt and deployed appropriate mathematical techniques

· learnt how to produce a dissertation

· acquired and practised skills of oral presentation

#### Year Four Optional Modules

##### Linear Differential Operators in Mathematical Physics (MATH421)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module provides a comprehensive introduction to the theory of partial differential equations, and it provides illustrative applications and practical examples in the theory of elliptic boundary value problems, wave propagation and diffusion problems.

**Learning Outcomes**(LO1) To understand and actively use the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green's functions.

(LO2) To apply powerful mathematical methods to problems of electromagnetism, elasticity, heat conduction and wave propagation.

(S1) Numeracy

##### Quantum Field Theory (MATH425)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of the essentials of quantum field theory.

**Learning Outcomes**(LO1) After the course the students should understand the important features of the mathematical tools necessary for particle physics. In particular they should · be able to compute simple Feynman diagrams, · understand the basic principles of regularisation and renormalisation · be able to calculate elementary scattering cross-sections.

##### Variational Calculus and Its Applications (MATH430)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**This module provides a comprehensive introduction to the theory of the calculus of variations, providing illuminating applications and examples along the way.

**Learning Outcomes**(LO1) Students will posses a solid understanding of the fundamentals of variational calculus

(LO2) Students will be confident in their ability to apply the calculus of variations to range of physical problems

(LO3) Students will also have the ability to solve a wide class of non-physical problems using variational methods

(LO4) Students will develop an understanding of Hamiltonian formalism and have the ability to apply this framework to solve physical and non-physical problems

(LO5) Students will be confident in their ability to analyse variational symmetries and generate the associated conservation laws

(S1) Problem solving skills

(S2) Numeracy

##### Classical Mechanics (PHYS470)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To provide students with an awareness of the physical principles that can be applied to understand important features of classical (i.e. non-quantum) mechanical systems. To provide students with techniques that can be applied to derive and solve the equations of motion for various types of classical mechanical systems, including systems of particles and fields. To develop students' understanding of the fundamental relationship between symmetries and conserved quantities in physics. To reinforce students’ knowledge of quantum mechanics, by developing and exploring the application of closely-related concepts in classical mechanics.

**Learning Outcomes**(LO1) Students should know the physical principles underlying the Lagrangian and Hamiltonian formulations of classical mechanics, in particular Newton's laws of motion and Hamilton’s principle, and should be able to explain the significance of Hamilton's principles in classical and modern physics.

(LO2) Students should be able to apply the Euler-Lagrange equations and Hamilton’s equations (as appropriate) to derive the equations of motion for specific dynamical systems, including complex nonlinear systems.

(LO3) Students should be able to use advanced concepts in classical mechanics to describe the connection between symmetries and conservation laws.

(LO4) Students should be able to apply advanced techniques, including conservation laws, canonical transformations, generating functions, perturbation theory etc. to describe important features of various dynamical systems (including systems of particles and fields) and to solve the equations of motion in specific cases.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

##### Accelerator Physics (PHYS481)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**60:40 **Aims**To build on modules on electricity, magnetism and waves;

To study the functional principle of different types of particle accelerators and their science and societal applications;

To study the generation of ion and electron beams;

To study the layout and the design of simple ion and electron optics;

To study basic concepts in radio frequency engineering and technology;

To understand the motion of beams of charged particles and their control

**Learning Outcomes**(LO1) At the end of the module the student should have: An understanding of the description of the motion of charged particles in complex electromagnetic fields; An understanding of different types of accelerators, in which energy range and for which purposes they are utilised; An understanding of the generation and technical exploitation of synchrotron radiation; An understanding of the concept and the necessity of beam cooling.

(S1) Presentation of recent research results in accelerator R&D through a scientific poster; learning about a new area through group discussions

##### Correlated Electron Materials (PHYS486)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**Students will develop an understanding of the physical properties that emerge due to electron-electron interactions and band structure in solids, including a critical awareness of properties that cannot be explained by treating electrons individually. They will be able to conceptually understand the theoretical emergence of, and parallels between, ferromagnetism, superconductivity and exemplar contemporary strongly correlated electron effects. Using these theories, students will be able to design experimental methods to measure such phenomena.

**Learning Outcomes**(LO1) Evaluate the impact of electron-electron interactions on the physical properties of matter

(LO2) Determine the mechanisms underpinning magnetic order and conventional superconductivity in solids, and their interrelation with key physical parameters

(LO3) Analyse experimental data from correlated electron phenomena using appropriate theoretical models

(LO4) Plan suitable experiments/methods to measure correlated electron phenomena

##### Physical Principles of Materials (PHYS487)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**To provide a science-led understanding of how materials are configured and how they behave. Fundamental concepts surrounding crystalline materials and their defects will be used to explain materials properties and behaviour, and frameworks for describing how materials respond to processing will be described. There will be exemplar case studies.

**Learning Outcomes**(LO1) Understand the types of point defects in solids and the thermodynamic drivers for their presence

(LO2) Understand the types of point defects in solids and the thermodynamic drivers for their presence

(LO3) Be able to use phase diagrams to predict the phase, composition and microstructure of solids.

(LO4) Know the Phase Rule and be able to apply it.

(LO5) Understand diffusion phenomena and be able to solve problems using Fick’s first and second laws. Understand the Kirkendall effect

(LO6) Have an appreciation of the fundamental drivers for crystal growth including supersaturation, nucleation and heat flow criteria.

(LO7) Know the main properties of dislocations in solids and how they influence the properties of materials, including mechanical properties. Understand diffusionless transformations

(LO8) Know about key materials classes through case studies.

##### Magnetic Structure and Function (PHYS497)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**p.MsoNormal, li.MsoNormal, div.MsoNormal{margin-top:0cm;margin-right:0cm;margin-bottom:8.0pt;margin-left:0cm;line-height:107%;font-size:11.0pt;font-family:"Calibri",sans-serif;}.MsoChpDefault{font-size:11.0pt;font-family:"Calibri",sans-serif;}.MsoPapDefault{margin-bottom:8.0pt;line-height:107%;}@page WordSection1{size:612.0pt 792.0pt;margin:72.0pt 72.0pt 72.0pt 72.0pt;}div.WordSection1{page:WordSection1;} To build on the third year module Condensed Matter Physics To develop an understanding of the phenomena and fundamental mechanisms of magnetism in condensed matter

**Learning Outcomes**(LO1) Have a basic understanding of the quantum origin of magnetism and magnetic moments.

(LO2) Understand the concept of magnetic order and the role of exchange interactions.

(LO3) Be able to identify the properties associated with various types of magnetism.

(LO4) Be able to explain the cause of magnetic phenomena such as hysteresis and domain formation.

##### Nanoscale Physics and Technology (PHYS499)

**Level**M **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**• To introduce the emerging fields of nanoscale physics and nanotechnology

• To describe experimental techniques for probing physical properties of nanostructured materials

• To describe the novel size-dependent electronic, optical, magnetic and chemical properties of nanoscale materials

• To describe several ‘hot topics' in nanoscience research

• To develop students' problem-solving, investigative, communication and analytic skills through appropriate assignments for tutorials and a literature project.**Learning Outcomes**(LO1) The ability to explain how and why nanoscale systems form.

(LO2) The ability to describe how nanoscale systems may be probed experimentally and compare different techniques in terms of strengths and weaknesses.

(LO3) The ability to explain and apply the fundamental principles that govern nanoscale systems.

(LO4) The ability to describe potential applications and to discuss their wider applications.

(S1) Critical thinking and problem solving - Critical analysis

(S2) Communication (oral, written and visual) - Presentation skills – oral

(S3) Communication (oral, written and visual) - Report writing

(S4) Critical thinking and problem solving - Evaluation

##### Introduction to String Theory (MATH423)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of string theory, and its utilization as a theory that unifies all of the known fundamental matter and interactions.

**Learning Outcomes**(LO1) After completing the module the students should: - be familiar with the properties of the classical string.

(LO2) be familiar with the basic structure of modern particle physics and how it may arise from string theory.

(LO3) be familiar with the basic properties of first quantized string and the implications for space-time dimensions.

(LO4) be familiar with string toroidal compactifications and T-duality.

(S1) Problem solving skills

(S2) Numeracy

##### Advanced Topics in Mathematical Biology (MATH426)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To introduce some hot problems of contemporary mathematical biology, including analysis of developmental processes, networks and biological mechanics.

To further develop mathematical skills in the areas of difference equations and ordinary and partial differential equations.

To explore biological applications of fluid dynamics in the limit of low and high Reynolds number.

**Learning Outcomes**(LO1) To familiarise with mathematical modelling methodology used in contemporary mathematical biology.

(LO2) Be able to use techniques from difference equations and ordinary and partial differential equations in tackling problems in biology.

##### Waves, Mathematical Modelling (MATH427)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**This module gives an introduction to the mathematical theory of linear and non-linear waves. Illustrative applications involve problems of acoustics, gas dynamics and examples of solitary waves.

**Learning Outcomes**(LO1) To understand essential modelling techniques in problems of wave propagation.

(LO2) To understand that mathematical models of the same type can be successfully used to describe different physical phenomena.

(LO3) To understand background mathematical theory in models of acoustics, gas dynamics and water waves.

(S1) Problem solving skills

(S2) Numeracy

##### Asymptotic Methods for Differential Equations (MATH433)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**This module provides an introduction into the perturbation theory for partial differential equations. We consider singularly and regularly perturbed problems and applications in electro-magnetism, elasticity, heat conduction and propagation of waves.

**Learning Outcomes**(LO1) The ability to make appropriate use of asymptotic approximations.

(LO2) The ability to analyse boundary layer effects.

(LO3) The ability to use the method of compound asymptotic expansions in the analysis of singularly perturbed problems.

(S1) Problem solving skills

(S2) Numeracy

##### Physics of Life (PHYS482)

**Level**M **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To introduce students to the physical principles needed to address important problems such as climate change, the loss of biodiversity, the understanding of ecological systems, the growth of resistance to antibiotics, the challenge of sustainable development and the study of disease. These problems offer excellent opportunities for rewarding careers.

**Learning Outcomes**(LO1) An understanding of the conditions necessary for life to evolve in a universe.

(LO2) An understanding of the thermodynamics and organization of living things.

(LO3) Familiarity with physical techniques used in the study of biological systems.

(LO4) An understanding of current ideas of how life may have evolved on earth.

(LO5) An understanding of how the earth’s climate has varied over geological time.

(S1) Problem solving skills.

##### Advanced Nuclear Physics (PHYS490)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To build on the year 3 modules on Nuclear Physics To offer an insight into current ideas about the description of atomic nuclei and nuclear matter

**Learning Outcomes**(LO1) Knowledge of the basic properties of nuclear forces and the experimental evidence upon which these are based

(LO2) Knowledge of the factors governing nuclear shapes

(LO3) Understanding of the origin of pairing forces and the effect of these and rotational forces on nuclear behaviour

(LO4) An overview of phenomena observed for exotic nuclei far from the line of nuclear stability

(LO5) Knowledge of astrophysical nucleosynthesis processes

(LO6) Knowledge of phases of nuclear matter

##### Neutrinos and Dark Matter (PHYS492)

**Level**M **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To build on PHYS377 to provide an understanding of neutrino physics and dark matter, including key experimental methods used in their detection. To provide an understanding of the low background experimental techniques that underpin both areas

**Learning Outcomes**(LO1) Understand neutrino physics including spin, flavour, neutrino oscillations, sterile neutrinos, neutrinoless double beta decay, dirac and majorana neutrinos, leptogenisis and cosmic neutrinos

(LO2) Understand dark matter including evidence, DM models including WMPS and axions, direct detection, indirect detection, DM detection at colliders

(LO3) Understand low background experimental techniques including, underground laboratories, cosmic muons, Th chains and K, Radon, Spallation and activation, neutrons

##### Advanced Particle Physics (PHYS493)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**To build on the Year 3 module PHYS377 Particle Physics to give the student a deeper understanding of the Standard Model of Particle Physics and the basic extensions to review the detectors and accelerator technology available to investigate the questions posed by the Standard Model and its extensions.

**Learning Outcomes**(LO1) An understanding of the Standard Model and its extensions. This will be placed in context of the understanding of the origin of the universe, its properties and its physical laws

(LO2) An understanding of how present and future detector and accelerator technology will be applied to investigate the development of the Standard Model

(LO2) An understanding of how present and future detector and accelerator technology will be applied to investigate the development of the Standard Model

(LO3) An understanding of the effects of symmetries on particle properties

(LO4) Ablity to caclulate decay rates for particles

(S1) Problem solving skills

(S2) International awareness

(S3) Organisational skills

(S4) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

The programme detail and modules listed are illustrative only and subject to change.

#### Teaching and Learning

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.

#### 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.