# Theoretical Physics MPhys

- Course length: 4 years
- UCAS code: F344
- Year of entry: 2019
- Typical offer: A-level : AAB / IB : 35 / BTEC : Applications considered

## 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

#### Year One Compulsory Modules

##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **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**differentiate and integrate a wide range of functions;

sketch graphs and solve problems involving optimisation and mensuration

understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

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

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **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**manipulate complex numbers and solve simple equations involving them

solve arbitrary systems of linear equations

understand and use matrix arithmetic, including the computation of matrix inverses

compute and use determinants

understand and use vector methods in the geometry of 2 and 3 dimensions

calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics

##### Thermal Physics (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 gasses
- Entropy
- The equation of state
- Van der Waals equation
- States of matter and state changes
- The basis of statistical mechanics

**Learning Outcomes**Construct a temperature scale and understand how to calibrate a thermometer with that scale

Calculate the heat flow into and work done by a system and how that is constrained by the first law of Thermodynamics

Analyse the expected performance of heat engines, heat pumps and refrigerators

Relate the second law of thermodynamics to the operation of heat engines, particularly the Carnot engine

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

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

Calculate the linear and volume thermal expansions of materials

Understand the basis of entropy and relate this to the second law of thermodynamics andcalculate entropy changesRelate the equation of state for a material to the macroscopic properties of the material

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

Be able to link the microscopic view of a system to its macroscopic state variablesBe able to demonstrate the equivalence of the Clausius and Kelvin-Planck statements of the second law of thermodynamics.

Be able to derive and use Maxwell''s equations

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

**Level**1 **Credit level**7.5 **Semester**First Semester **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**Ability to produce algorithms to solve simple physical problems.

Ability to program and use simple algorithms on a computer

Ability to analyse and present physical data

Ability to produce simple Monte Carlo models

Ability to carry out basic symbolic manipulations using a computer

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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**use Taylor series to obtain local approximations to functions;

obtain partial derivaties and use them in several applications such as, error analysis, stationary points change of variables

evaluate double integrals using Cartesian and Polar Co-ordinates

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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**

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

oscillation, vibration, resonance

##### 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**Demonstrate an understanding of oscillators.

Understand the fundamental principles underlying wave phenomena.

Apply those principles to diverse phenomena.

Understand wave reflection and transmission, superposition of waves.

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

Understand linear and circular polarisation.

Understand inteference and diffraction effects.

Understand lenses and optical instruments.

Apply Fourier techniques and understand their link to diffraction patterns.

Understand the basic principles of lasers

##### Foundations of Modern Physics (PHYS104)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**60:40 **Aims**- To introduce the theory of special relativity and its experimental proofs.
- To carry out calculations using relativity and visualise them.
- 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 relativity and quantum theory on contemporary science and society.

**Learning Outcomes**An understanding why classical mechanics must have failed to describe the properties of light, the motion of objects with speeds close to the speed of light and the properties of microspopic systems.

A basic knowledge on the experimental and theoretical concepts which founded modern physics, i.e. that either relativity or quantum theory or both are needed to explain certain phenomena.

A knowledge of the postulates of special relativity.

An understanding of the concept of spacetime, of the relativity of length, time and velocity.

An ability to apply the Lorentz transformation and the concept of Lorentz invariance to simple cases

An ability to apply the equations of relativistic energy, momentum and rest mass.

An understanding of the Doppler effect for light and visualisation of relativistic effects.

An ability to solve problems based on special relativity.

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

An understanding of the quantum theory of light and the ability to apply the energy-momentum conservation for light, e.g. photo-electric effect, Compton effect.

An understanding of the structure of atoms and its experimental foundations.

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

An understanding of de Broglie waves and their statistical interpretation.

An ability to explain the experimental evidence of de Broglie waves with scattering experiments of electrons, X-rays and neutrons.

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

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

A basic knowledge of contemporary applications of quantum theory and relativity, e.g. nuclear reactor and nuclear fissions, and the impact on our society.

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

**Level**1 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**- 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**Appreciation of the practical nature of physics

Awareness of the importance of accurate experimentation, particularly obervation and record keeping

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

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

Problem solving skills of a practical nature

Analytical skills in the analysis of the data

Investgative skills in performing the experiment and extracting information from various sources with which to compare the results

Ability to organise their time and meet deadlines

### Programme Year Two

In the second and subsequent years of all programmes, there is a wide range of modules. For the programme that you choose there may be no compulsory modules (although you may have to choose a few from a subset such as Pure Mathematics). If you make a different choice, you will find that one or more modules have to be taken. Each year you will choose the equivalent of eight modules. Please note that we regularly review our teaching so the choice of modules may change.

#### Year Two Compulsory Modules

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

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**85:15 **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**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.

##### Complex Functions (MATH243)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**To introduce the student to a surprising, very beautiful theory which has intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

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

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

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

To understand various Cauchy formulae and theorems and their applications.

To be able to reduce a real definite integral to a contour integral.

To be competent at computing contour integrals.

##### Electromagnetism (PHYS201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**- To introduce the fundamental concepts and principles of electrostatics, magnetostatics, electromagnetism and 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**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.

Apply differential vector analysis to electromagnetism.

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.

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.

##### Condensed Matter Physics (PHYS202)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims** The aims of Phys202 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 course is to introduce its central ideas and methodology to the students.

Condensed matter refers to both liquids and solids and all kinds of other forms of matter in between those two extremes, generally known as “soft matter". While the course will touch on liquids, the emphasis will be on crystalline solids, including some nano-materials. The reason for focusing on crystals is that the periodicity of a crystal is what allows us to make progress in developing a theory for various phenomena in solids based on first principles. Two important concepts are:

• the electronic states of electrons in a solid and

• the vibrations of atoms in the solid.

The description of these ideas basically refer to the theory of electronic band structure and the theory of phonons. These concepts form the basis for understanding a wide range of phenomena including how the atoms bond together to form the crystal, what are some basic statistical properties like specific heat, how electrons move in solids and electronic transport, why are some materials metals and others semiconductors and insulators, and how do solids interact with electromagnetic fields. The course will also introduce optical and magnetic properties in solids, scattering phenomena, thermal conductivity and effect of defects in solids, semiconductors, magnetism and go beyond the free electron model to touch on intriguing effects such as superconductivity.

**Learning Outcomes**On satisfying the requirements of this course, 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.

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

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

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

Students will understand the concept of electron and hole carrier statistics, effective masses and transport in intrinsic and extrinsic semiconductorsStudents 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.

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.

##### Introduction to the Methods of Applied Mathematics (MATH224)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics.

To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics.

**Learning Outcomes**After completing the module students should:

- be fluent in the solution of basic ordinary differential equations, including systems of first order equations;

- be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations;

- be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations;

- be able to solve simple first order partial differential equations;

- be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.

##### Classical Mechanics (MATH228)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide an understanding of the principles of Classical Mechanics and their application to dynamical systems.

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

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

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

To understand the connection between symmetry and conservation laws.

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

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

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **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**At the end of the module the student should have:

- An understanding of the reasons why microscopic systems require quantum description and statistical interpretation.
- Knowledge of the Schrodinger equation and how it is formulated to describe simple physical systems.
- Understanding of the basic technique of using Schrodinger''s equation and ability to determine solutions in simple cases.
- Understanding of how orbital angular momentum is described in quantum mechanics and why there is a need for spin.
- Understanding how the formalism of quantum mechanics describes the structure of atomic hydrogen and, schematically, how more complex atoms are described.

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

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **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**basic understanding of Rutherford, electron on neutron scattering

understanding of the basic principles that determine nuclear size, mass and decay modes

knowledge of examples and applications of nuclear physics

knowledge of elementary particles and their interactionsbasic understanding of relativistic 4-vectors

### Programme Year Three

Choose three modules from:

- Quantum Mechanics (MATH325) OR PHYS361 Quantum Mechanics & Atomic Physics
- Relativity (MATH326)
- Mathematical Physics Project (MATH432) or Modelling Physical Phenomena (Project) (PHYS488)

Choose at least three modules from:

- Further Methods of Applied Mathematics (MATH323)
- Cartesian Tensors and Mathematical Models of Solids and Viscous Fluids (MATH324)
- Population Dynamics (MATH332)
- Condensed Matter Physics (PHYS363)
- Nuclear Physics (PHYS375)
- Practical Physics III (PHYS306)
- Materials Physics (PHYS387)
- Semiconductor Applications (PHYS389)
- Communicating Science (PHYS391)
- Statistics in Data Analysis (PHYS392)
- Statistical and Low Temperature Physics (PHYS393)
- Mathematical Economics (MATH331)
- Chaos and Dynamical Systems (MATH322)
- Riemann Surfaces (MATH340)
- The Magic of Complex Numbers: Complex Dynamics, Chaos and the Madelbrot Set (MATH345)
- Differential Geometry (MATH349)
- Advanced Electromagnetism (PHYS370)
- Relativity and Cosmology (PHYS374)
- Introduction to Particle Physics (PHYS377)
- Surface Physics (PHYS381)
- Physics of Life (PHYS382)
- Physics of Energy Sources (PHYS388)
- Undergraduate Ambassadors Project (PHYS 396)
- Technology Transfer and Commercialisation (PHYS 397)

Choose at least two from;

- Linear Differential Operators in Mathematical Physics (MATH421)
- Quantum Field Theory (only in Year 4) (MATH425)
- Variational Calculus and its Applications (MATH430)
- Classical Mechanics (PHYS470)
- Accelerator Physics (PHYS481)
- Research Skills (PHYS491)
- Nanoscale Physics and Technology (PHYS499)
- Magnetic Structure and Function (PHYS497)
- Introduction to String Theory (MATH423)
- Analytical and Computational Methods for Applied Mathematics (MATH424)
- Advanced Topics in Mathematical Biology (MATH426)
- Waves, Mathematical Modelling (MATH427)
- Introduction to Modern Particle Theory (MATH431)
- Asymptotic Methods for Differential Equations (MATH433)
- Advanced Nuclear Physics (PHYS490)
Advanced Particle Physics (PHYS493)

#### Year Three Optional Modules

##### Statistical and Low Temperature Physics (PHYS393)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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**Understanding of the statistical basis of entropy and temperature

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

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

Knowledge of cooling techniques

Knowledge and understanding of basic theories of liquid helium behaviour and superconductivity in cooling techniques

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing the module students should:

· Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

· Be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

· Be able to solve mathematically a variety of standard problems in the theory of games.

· To understand the relevance of such solutions in real situations.

##### Chaos and Dynamical Systems (MATH322)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To develop expertise in dynamical systems in general and study particular systems in detail.

**Learning Outcomes**After completing the module students will be able to understand the possible behaviour of dynamical systems with particular attention to chaotic motion;

After completing the module students will be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points;

After completing the module students will understand how fractal sets arise and how to characterise them.

##### Riemann Surfaces (MATH340)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.

**Learning Outcomes**Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **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'' understanding of formal reasoning and proof. **Learning Outcomes**To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

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

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

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

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

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

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **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**1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces. 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

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.

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

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

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

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

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

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

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

##### Advanced Electromagnetism (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.
- To further develop students'' problem-solving and analytic skills.

**Learning Outcomes**Students should have 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).

Students should have an understanding of the behaviour of electromagnetic waves at boundaries.

Students should have an understanding of the behaviour of electromagnetic waves in cavities, waveguides and transmission lines.

Students should have an understanding of the properties of electric dipole radiation.

Students should have the ability to explain an explicity covariant formulation of electromagnetism in special relativity.

##### Relativity and Cosmology (PHYS374)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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**The ability to explain the relationship between Newtonian gravity and Einstein''s General Relativity (GR) Understanding of the concept of curved space time and knowledge of metrics.

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

Knowledge of how simple cosmological models of the universe are constructed.

The ability to calculate physical parameters and make observational predictions for a range of such models.##### Introduction to Particle Physics (PHYS377)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To build on the second year module involving Nuclear and Particle Physics
- To develop an understanding of the modern view of particles, of their interactions and the Standard Model

**Learning Outcomes**At the end of the module the student should have:

Basic understanding of relativistic kinematics (as applied to collisions, decay processes and cross sections)

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

Knowledge of the fundamental particles of the Standard Model and the experimental evidence for the Standard Model

Knowledge of conservation laws and discrete symmetries

##### Surface Physics (PHYS381)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- Develop a syllabus to describe the properties of surfaces
- Convey an understanding of the physical properties of Surfaces
- Provide knowledge of a raneg of surface characterisation techniques
- Illustrate surface processes and their relevance to technologies

**Learning Outcomes**explain how the presence of the surface alters physical properties such as atomic an electronic structure

choose the right characterisation technique to assess different surface properties have gained an appreciation of surface processes and their relevance to the modification of surface propertiesbe able to describe surface alterations and processes using the right terminology

##### Physics of Life (PHYS382)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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** An understanding of the conditions necessary for life to evolve in a universe.

An understanding of the thermodynamics and organization of living things.

Familiarity with physical techniques used in the study of biological systems. ##### Physics of Energy Sources (PHYS388)

**Level**3 **Credit level**15 **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**At the end of the module the student should have:

- Learned the fundamental physical principles underlying energy production using conventional and renewable energy sources
- Learned the fundamental physical principles underlying nuclear fission and fusion reactors
- Studied the applications of these principles in the design issues power generation
- An appreciation of the role of mathematics in modelling power generation
- Learned the fundamental physical principles concerning the origin and consequences of environmental radioactivity
- Developed an awareness of the safety issues involved in exposure to radiation
- Developed problem solving skills based on the material presented
- Developed an appreciation of the problems of supplying the required future energy needs and the scope and issues associated with the different possible methods

##### Undergraduate Ambassadors Project (PHYS396)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**- To provide undergraduates with key transferable skills.
- To provide students with opportunity to learn to communicate physics at different levels.
- To provide students with work-place experience.
- To provide students with the opportunity to work with staff in a different environment with different priorities to the University.
- To provide teaching experience that encourages undergraduates to consider a career in teaching.
- To supply role models for secondary school students.
- To provide support and teaching assistance to secondary school teachers.
- To encourage a new generation of physicists.

**Learning Outcomes**Communicate physicseffectively to others

Plan a lesson

Design a worksheet

Evaluate their planning

Assess the effectiveness of a session or worksheet that they have designed

Manage small groups ofpupils (e.g. to complete an experiment)

Prioritise their work

##### Technology Transfer and Commercialisation (PHYS397)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**This module aims to

- To be able to develop skills in assessing thecommercial routes available to introduce a product or service into the market.

- To be adept in market information gathering andanalysis.

- To develop presentation and communicationskills and reporting skills beyond the classic essay format.

- To distinguish clearly between thedifferent business models available and to contrast merits and drawbacks ofeach solution.

**Learning Outcomes**All students will be able to gather and analyse business data information

All students will be able to understand technology transfer dynamics

students will be able to communicate their ideas and work in a clear and concise mannerStudents will be able to present data and project proposals in a professional manner, easily recognised by industry and companies.

##### 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**To understand and actively use the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green''s functions.

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

##### 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**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**90:10 **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**Students will posses a solid understanding of the fundamentals of variational calculus

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

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

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

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

##### Classical Mechanics (PHYS470)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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**Students should know the physical principles underlying the Lagrangian and Hamiltonian formulations of classical mechanics, in particular D’Alembert’s principle and Hamilton’s principle, and should be able to explain the significance of these advanced principles in classical and modern physics.

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.

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

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.

##### Accelerator Physics (PHYS481)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**- To build on modules on electricity, magnetism and waves;
- To study the functional principle of different types of particle accelerators;
- 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.

**Learning Outcomes**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.

##### Research Skils (PHYS491)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**- Perform literature searches.
- Plan research projects.
- Explain research projects to both expert and non-expert audiences.
- Organise a team of people and work as a group.
- Assess the broader impact of research projects.
- Present a proposal as a written document ans orally.

**Learning Outcomes**Experience in carrying out search of scientific literature. Communicating research to non-expert audience.

Evaluating the possible broader impact of research.Writing a scientific case for an assessment panel. First experience with some project management tools.

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

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **Aims**- Tointroduce the emerging fields of nanoscale physics and nanotechnology
To describe experimental techniques for probing physical properties of nanostructured materials

Todescribe the novel size-dependent electronic, optical, magnetic and chemicalproperties of nanoscale materials

Todescribe several ‘hot topics'' in nanoscience research

Todevelop students'' problem-solving, investigative, communication and analyticskills through appropriate assignments for tutorials and a literature project.

**Learning Outcomes**After the module the students should have the ability to explain how and why nanoscalesystems form.

After the module the students should have the ability to describe how nanoscale systems may be probed experimentally and compare different techniques in terms of strengths and weaknesses.After the module the students should have the ability to explain and apply the fundamental principles that govern nanoscale systems.

After the module the students should have the ability to describe potential applications and to discuss their wider applications.

After the module the students should have enhanced problem-solving, investigative, communication, and analytic skills.

- Tointroduce the emerging fields of nanoscale physics and nanotechnology
##### Magnetic Structure and Function (PHYS497)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**- 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**Have a basic understanding of the quantum origin of magnetism and magnetic moments. Understand the concept of magnetic order and the role of exchange interactions.Be able to identify the properties associated with various types of magnetism.

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

##### 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**After completing the module the students should:

- be familiar with the properties of the classical string.

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

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

be familiar with string toroidal compactifications and T-duality.##### Analytical & Computational Methods for Applied Mathematics (MATH424)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to a range of analytical and numerical methods for partial differential equations arising in many areas of applied mathematics.

To provide a focus on advanced analytical techniques for solution of both elliptic and parabolic partial differential equations, and then on numerical discretisation methods of finite differences and finite elements.

To provide the algorithms for solving the linear equations arising from the above discretisation techniques.

**Learning Outcomes**Apply a range of standard numerical methods for solution of PDEs and should have an understanding of relevant practical issues.

Obtain solutions to certain important PDEs using a variety of analytical techniques and should be familiar with important properties of the solution.

Understand and be able to apply standard approaches for the numerical solution of linear equations

Have a basic understanding of the variation approach to inverse problems.

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

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**To familiarise with mathematical modelling methodology used in contemporary mathematical biology. 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**100:0 **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**To understand essential modelling techniques in problems of wave propagation.

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

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

##### Introduction to Modern Particle Theory (MATH431)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of the current status of elementary particle theory.

To describe the structure of the Standard Model of particle physics and its embedding in Grand Unified Theories.

**Learning Outcomes**To understand the Lorentz and Poincare groups and their role in classification of elementary particles.

To understand the basics of Langrangian and Hamiltonian dynamics and the differential equations of bosonic and fermionic wave functions.

To understand the basic elements of field quantisation.

To understand the Feynman diagram pictorial representation of particle interactions.To understand the role of symmetries and conservation laws in distinguishing the strong, weak and electromagnetic interactions.

To be able to describe the spectrum and interactions of elementary particles and their embedding into Grand Unified Theories (GUTs)

To understand the flavour structure of the standard particle model and generation of mass through symmetry breaking.

To understand the phenomenological aspects of Grand Unified Theories.

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

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**The ability to make appropriate use of asymptotic approximations.

The ability to analyse boundary layer effects.

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

##### 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**Knowledge of the basic properties of nuclear forces and the experimental evidence upon which these are based

Knowledge of the factors governing nuclear shapes

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

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

Knowledge of astrophysical nucleosynthesis processes

Knowledge of phases of nuclear matter

##### Advanced Particle Physics (PHYS493)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**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

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

An understanding of the effects of symmetries on particle properties

Ablity to caclulate decay rates for particles

### 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.

In addition to compuilsory modules, choose at least three modules from:

- Further Methods of Applied Mathematics (MATH323)
- Cartesian Tensors and Mathematical Models of Solids and Viscous Fluids (MATH324)
- Population Dynamics (MATH332)
- Condensed Matter Physics (PHYS363)
- Nuclear Physics (PHYS375)
- Practical Physics III (PHYS306)
- Materials Physics (PHYS387)
- Semiconductor Applications (PHYS389)
- Communicating Science (PHYS391)
- Statistics in Data Analysis (PHYS392)
- Statistical and Low Temperature Physics (PHYS393)
- Mathematical Economics (MATH331)
- Chaos and Dynamical Systems (MATH322)
- Riemann Surfaces (MATH340)
- The Magic of Complex Numbers: Complex Dynamics, Chaos and the Madelbrot Set (MATH345)
- Differential Geometry (MATH349)
- Advanced Electromagnetism (PHYS370)
- Relativity and Cosmology (PHYS374)
- Introduction to Particle Physics (PHYS377)
- Surface Physics (PHYS381)
- Physics of Life (PHYS382)
- Physics of Energy Sources (PHYS388)
- Undergraduate Ambassadors Project (PHYS396)
- Technology Transfer and Commercialisation (PHYS397)

Choose at least two from:

- Linear Differential Operators in Mathematical Physics (MATH421)
- Quantum Field Theory (only in Year 4) (MATH425)
- Variational Calculus and its Applications (MATH430)
- Classical Mechanics (PHYS470)
- Accelerator Physics (PHYS481)
- Research Skills (PHYS491)
- Nanoscale Physics and Technology (PHYS499)
- Magnetic Structure and Function (PHYS497)
- Introduction to String Theory (MATH423)
- Analytical and Computational Methods for Applied Mathematics (MATH424)
- Advanced Topics in Mathematical Biology (MATH426)
- Waves, Mathematical Modelling (MATH427)
- Introduction to Modern Particle Theory (MATH431)
- Asymptotic Methods for Differential Equations (MATH433)
- Advanced Nuclear Physics (PHYS490)
- Advanced Particle Physics (PHYS493)

#### Year Four Compulsory Modules

##### Advanced Quantum Physics (PHYS480)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**- To build on Y3 module on Quantum Mechanics and Atomic Physics 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**At the end of the module the student should have:

- Understanding of variational techniques.
- 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

Understanding of examples of state-of-the art quantum physics experiments.

Understanding of the implications of quantum physics in our daily lifes.

##### 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**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

##### Statistical and Low Temperature Physics (PHYS393)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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**Understanding of the statistical basis of entropy and temperature

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

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

Knowledge of cooling techniques

Knowledge and understanding of basic theories of liquid helium behaviour and superconductivity in cooling techniques

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing the module students should:

· Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

· Be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

· Be able to solve mathematically a variety of standard problems in the theory of games.

· To understand the relevance of such solutions in real situations.

##### Chaos and Dynamical Systems (MATH322)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To develop expertise in dynamical systems in general and study particular systems in detail.

**Learning Outcomes**After completing the module students will be able to understand the possible behaviour of dynamical systems with particular attention to chaotic motion;

After completing the module students will be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points;

After completing the module students will understand how fractal sets arise and how to characterise them.

##### Riemann Surfaces (MATH340)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.

**Learning Outcomes**Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **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'' understanding of formal reasoning and proof. **Learning Outcomes**To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

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

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

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

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

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

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **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**1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces. 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

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.

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

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

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

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

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

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

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

##### Advanced Electromagnetism (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.
- To further develop students'' problem-solving and analytic skills.

**Learning Outcomes**Students should have 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).

Students should have an understanding of the behaviour of electromagnetic waves at boundaries.

Students should have an understanding of the behaviour of electromagnetic waves in cavities, waveguides and transmission lines.

Students should have an understanding of the properties of electric dipole radiation.

Students should have the ability to explain an explicity covariant formulation of electromagnetism in special relativity.

##### Relativity and Cosmology (PHYS374)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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**The ability to explain the relationship between Newtonian gravity and Einstein''s General Relativity (GR) Understanding of the concept of curved space time and knowledge of metrics.

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

Knowledge of how simple cosmological models of the universe are constructed.

The ability to calculate physical parameters and make observational predictions for a range of such models.##### Introduction to Particle Physics (PHYS377)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To build on the second year module involving Nuclear and Particle Physics
- To develop an understanding of the modern view of particles, of their interactions and the Standard Model

**Learning Outcomes**At the end of the module the student should have:

Basic understanding of relativistic kinematics (as applied to collisions, decay processes and cross sections)

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

Knowledge of the fundamental particles of the Standard Model and the experimental evidence for the Standard Model

Knowledge of conservation laws and discrete symmetries

##### Surface Physics (PHYS381)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- Develop a syllabus to describe the properties of surfaces
- Convey an understanding of the physical properties of Surfaces
- Provide knowledge of a raneg of surface characterisation techniques
- Illustrate surface processes and their relevance to technologies

**Learning Outcomes**explain how the presence of the surface alters physical properties such as atomic an electronic structure

choose the right characterisation technique to assess different surface properties have gained an appreciation of surface processes and their relevance to the modification of surface propertiesbe able to describe surface alterations and processes using the right terminology

##### Physics of Life (PHYS382)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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** An understanding of the conditions necessary for life to evolve in a universe.

An understanding of the thermodynamics and organization of living things.

Familiarity with physical techniques used in the study of biological systems. ##### Physics of Energy Sources (PHYS388)

**Level**3 **Credit level**15 **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**At the end of the module the student should have:

- Learned the fundamental physical principles underlying energy production using conventional and renewable energy sources
- Learned the fundamental physical principles underlying nuclear fission and fusion reactors
- Studied the applications of these principles in the design issues power generation
- An appreciation of the role of mathematics in modelling power generation
- Learned the fundamental physical principles concerning the origin and consequences of environmental radioactivity
- Developed an awareness of the safety issues involved in exposure to radiation
- Developed problem solving skills based on the material presented
- Developed an appreciation of the problems of supplying the required future energy needs and the scope and issues associated with the different possible methods

##### Undergraduate Ambassadors Project (PHYS396)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**- To provide undergraduates with key transferable skills.
- To provide students with opportunity to learn to communicate physics at different levels.
- To provide students with work-place experience.
- To provide students with the opportunity to work with staff in a different environment with different priorities to the University.
- To provide teaching experience that encourages undergraduates to consider a career in teaching.
- To supply role models for secondary school students.
- To provide support and teaching assistance to secondary school teachers.
- To encourage a new generation of physicists.

**Learning Outcomes**Communicate physicseffectively to others

Plan a lesson

Design a worksheet

Evaluate their planning

Assess the effectiveness of a session or worksheet that they have designed

Manage small groups ofpupils (e.g. to complete an experiment)

Prioritise their work

##### Technology Transfer and Commercialisation (PHYS397)

**Level**3 **Credit level**7.5 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**This module aims to

- To be able to develop skills in assessing thecommercial routes available to introduce a product or service into the market.

- To be adept in market information gathering andanalysis.

- To develop presentation and communicationskills and reporting skills beyond the classic essay format.

- To distinguish clearly between thedifferent business models available and to contrast merits and drawbacks ofeach solution.

**Learning Outcomes**All students will be able to gather and analyse business data information

All students will be able to understand technology transfer dynamics

students will be able to communicate their ideas and work in a clear and concise mannerStudents will be able to present data and project proposals in a professional manner, easily recognised by industry and companies.

##### 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**To understand and actively use the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green''s functions.

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

##### 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**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**90:10 **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**Students will posses a solid understanding of the fundamentals of variational calculus

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

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

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

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

##### Classical Mechanics (PHYS470)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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**Students should know the physical principles underlying the Lagrangian and Hamiltonian formulations of classical mechanics, in particular D’Alembert’s principle and Hamilton’s principle, and should be able to explain the significance of these advanced principles in classical and modern physics.

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.

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

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.

##### Accelerator Physics (PHYS481)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**- To build on modules on electricity, magnetism and waves;
- To study the functional principle of different types of particle accelerators;
- 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.

**Learning Outcomes**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.

##### Research Skils (PHYS491)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**- Perform literature searches.
- Plan research projects.
- Explain research projects to both expert and non-expert audiences.
- Organise a team of people and work as a group.
- Assess the broader impact of research projects.
- Present a proposal as a written document ans orally.

**Learning Outcomes**Experience in carrying out search of scientific literature. Communicating research to non-expert audience.

Evaluating the possible broader impact of research.Writing a scientific case for an assessment panel. First experience with some project management tools.

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

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**70:30 **Aims**- Tointroduce the emerging fields of nanoscale physics and nanotechnology
To describe experimental techniques for probing physical properties of nanostructured materials

Todescribe the novel size-dependent electronic, optical, magnetic and chemicalproperties of nanoscale materials

Todescribe several ‘hot topics'' in nanoscience research

Todevelop students'' problem-solving, investigative, communication and analyticskills through appropriate assignments for tutorials and a literature project.

**Learning Outcomes**After the module the students should have the ability to explain how and why nanoscalesystems form.

After the module the students should have the ability to describe how nanoscale systems may be probed experimentally and compare different techniques in terms of strengths and weaknesses.After the module the students should have the ability to explain and apply the fundamental principles that govern nanoscale systems.

After the module the students should have the ability to describe potential applications and to discuss their wider applications.

After the module the students should have enhanced problem-solving, investigative, communication, and analytic skills.

- Tointroduce the emerging fields of nanoscale physics and nanotechnology
##### Magnetic Structure and Function (PHYS497)

**Level**M **Credit level**7.5 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**- 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**Have a basic understanding of the quantum origin of magnetism and magnetic moments. Understand the concept of magnetic order and the role of exchange interactions.Be able to identify the properties associated with various types of magnetism.

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

##### 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**After completing the module the students should:

- be familiar with the properties of the classical string.

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

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

be familiar with string toroidal compactifications and T-duality.##### Analytical & Computational Methods for Applied Mathematics (MATH424)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to a range of analytical and numerical methods for partial differential equations arising in many areas of applied mathematics.

To provide a focus on advanced analytical techniques for solution of both elliptic and parabolic partial differential equations, and then on numerical discretisation methods of finite differences and finite elements.

To provide the algorithms for solving the linear equations arising from the above discretisation techniques.

**Learning Outcomes**Apply a range of standard numerical methods for solution of PDEs and should have an understanding of relevant practical issues.

Obtain solutions to certain important PDEs using a variety of analytical techniques and should be familiar with important properties of the solution.

Understand and be able to apply standard approaches for the numerical solution of linear equations

Have a basic understanding of the variation approach to inverse problems.

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

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**To familiarise with mathematical modelling methodology used in contemporary mathematical biology. 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**100:0 **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**To understand essential modelling techniques in problems of wave propagation.

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

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

##### Introduction to Modern Particle Theory (MATH431)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of the current status of elementary particle theory.

To describe the structure of the Standard Model of particle physics and its embedding in Grand Unified Theories.

**Learning Outcomes**To understand the Lorentz and Poincare groups and their role in classification of elementary particles.

To understand the basics of Langrangian and Hamiltonian dynamics and the differential equations of bosonic and fermionic wave functions.

To understand the basic elements of field quantisation.

To understand the Feynman diagram pictorial representation of particle interactions.To understand the role of symmetries and conservation laws in distinguishing the strong, weak and electromagnetic interactions.

To be able to describe the spectrum and interactions of elementary particles and their embedding into Grand Unified Theories (GUTs)

To understand the flavour structure of the standard particle model and generation of mass through symmetry breaking.

To understand the phenomenological aspects of Grand Unified Theories.

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

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**The ability to make appropriate use of asymptotic approximations.

The ability to analyse boundary layer effects.

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

##### 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**Knowledge of the basic properties of nuclear forces and the experimental evidence upon which these are based

Knowledge of the factors governing nuclear shapes

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

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

Knowledge of astrophysical nucleosynthesis processes

Knowledge of phases of nuclear matter

##### Advanced Particle Physics (PHYS493)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**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

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

An understanding of the effects of symmetries on particle properties

Ablity to caclulate decay rates for particles

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.