Physics with Astrophysics BSc (Hons)

The module is designed to provide you with the advanced mathematical and statistical techniques required to underpin study of physics at level 5 and beyond. Techniques covered will include Matrices, Fourier Series and Fourier and Laplace Transforms, Probability distributions, and an introduction to vector calculus (including div, grad and curl).

Students will develop skills in the use of advanced mathematical and statistical techniques, applying suitable mathematical calculations over a range of key topics, including explaining how a periodic waveform can be represented as an infinite series of sinusoids, and applying Fourier Transforms. The concepts of the eigenvalue and eigenvectors of a matrix, and how these can be found by algebraic means will also be covered. Finally, students will be introduced to vector calculus and vector operators, including div, grad and curl, and the Kronecker delta and Levi-Civita epsilon.

Linear Algebra
Algebraic evaluation of the eigenvalues and eigenvectors of a matrix (i.e. Matrices to the level of eigenvalues and eigenvectors). Application to the solution of a system of linear ordinary differential equations.

Vector Calculus
Coordinate systems; line, surface and volume integrals; Vector operators Grad, Div and Curl; Gauss’ (Divergence) Theorem, Stokes’ Theorem; Introduction to Cartesian tensors. Applications of vector calculus.

Fourier Series and Fourier and Laplace Transforms
Fourier series and periodic functions. Full-range and half-range series. Even and odd functions. Coefficients in complex form. Application to the solution of partial differential equations by the method of separation of variables. Fourier Transforms. Laplace Transforms. The convolution theorem. An introduction to the solution of partial differential equations.

Probability Distributions
Sample space, types of events, definition of probability, addition and multiplication laws, conditional probability. Discrete probability distributions including Binomial, Poisson. Continuous probability distributions including the Normal distribution.

The theory of Electromagnetism and its relativistic foundation is at the heart of modern physics and provides a fundamental paradigm for understanding contemporary physical theories. This module will introduce you to the fundamental concepts of Special Relativity, and to the origins and properties of electric and magnetic fields. A research inquiry approach will bring you through the step-by-step processes that led to the discovery of the Maxwell equations and understanding their relativistic nature.

Electrostatics and Magnetostatics
Coulomb's law and the electric field; Electric flux and Gauss' law; Circulation and electric potential; Calculating the field from the potential (gradient); Gauss law in differential form (divergence); Circulation law in differential form (curl); Poisson's and Laplace's equations and their solutions. Polarization, multi-pole expansion, electric potential of a dipole.
Definition of magnetic field and calculation of the force; Calculating the B field: the Biot-Savart law; Circulation and Ampere's law in differential form; Magnetic flux and Gauss law in differential form; Magnetic vector potential. Equation of motion of a charge in a electromagnetic field; cycloid motion; cyclotron frequency.
Electrical and magnetic fields in materials; Electrostatic fields and conductors (method of images); Electrostatic fields in dielectrics; Magnetostatic fields in materials.

Electrodynamics
Maxwell’s Equations. Ohm’s Law in differential form; Electromotive force; Electromagnetic induction, Faraday's law in differential form; Ampere-Maxwell law in differential form; Maxwell's equations and their solutions.
Electromagnetic Waves. Derivation from Maxwell's Equations; speed of light; Energy flow, Poynting vector. Electromagnetic wave polarisation; Incident, reflected and transmitted waves at plane interfaces.

Special Relativity
Einstein's postulates; Time dilation; Length contraction; Lorentz transformations; Light cone; Relativistic velocity transformation, energy-momentum relation; Twin paradox. Relativistic invariance of charge; Lorentz transformation of electromagnetic fields; EM of moving charges.

This module aims to equip physics students with the knowledge and transferable skills involved in computational methods and experimental techniques. Students will analyse and present experimental data, create computational models for appropriate physical systems and perform comparisons between theory and experiment. Quantitative, analytical and modelling training acquired in this module will support students’ professional and personal skills. This module offers the additional opportunity of research-orientated learning through a hands-on approach to analysing research-based data.

Experiments - Topics may include (note this is indicative rather than prescriptive):
1. Doppler Effect
2. Optical properties of semiconductors
3. Particle accumulation on a glass surface (c.f. sand particles on photovoltaic modules and link to Monte Carlo)
4. The heat engine
5. Hall Effect
6. Fundamentals properties of X-rays
7. Radioactive decay of ?, ? and ? particles
8. Microwave Diffraction
9. PID Control
10. Thermal Conductivity
11. Cosmic Ray Detection
12. Solar photovoltaic efficiency measurement.

Computation - Topics may include (note this is indicative rather than prescriptive):
1. Curve fitting (linear and non-linear), statistical analysis and data presentation
2. Matrices to the level of eigenvectors and eigenvalues
3. Discretisation and series analysis
4. Ordinary differential equations
5. Partial differential equations (links to stock market modelling, radioactivity, electrical and mechanical systems)
6. Thermal modelling
7. Probability distribution functions

At very small scales, classical mechanics (Newton’s laws) breaks down and quantum mechanics must be used. This module introduces the foundations of quantum mechanics starting from the failure of classical physics to describe important experiments and the concept of wave-particle duality. Students are then introduced to the concept of a particle’s wave function and solving the Schrödinger equation for standard problems.

Key parts of quantum mechanics that are covered within the module include:

The Origins of Quantum Mechanics
Bohr model of the atom. Quantised nature of light and atomic spectra. Failure of classical mechanics to describe key experiments. The photoelectric effect. Young’s double slit experiment. Wave nature of particles. Concept of wave function and localisation. De Broglie equation. Heisenberg Uncertainty Principle. Quantum numbers and Pauli Exclusion Principle.

The Schrödinger Equation and Standard Solutions
Time dependent Schrödinger equation and general formulation. Wave function normalisation. Time independent Schrödinger equation. Boundary conditions. Infinite square well. Finite square well. Tunnelling through a potential barrier. Harmonic Oscillator. Three dimensional Schrödinger equation. Particle in a box. Hydrogen atom.

Matrix Mechanics
Postulates of quantum mechanics. Operators and representation of dynamical variables. Eigenfunctions and eigenvalues and linear combinations. Hamiltonian and operator representation of the Schrödinger equation. Hermitian operators. Expectation values. Commutating operators. Harmonic oscillator: raising and lowering operators. Angular momentum and spin. Time independent perturbation theory

Particle Physics
Fundamental Forces. Particle Classification and the Standard Model. Particle interactions, reactions and decays.

The module is aimed at providing students with core knowledge and understanding in semiconductor physics. The module also gives students an opportunity to develop professional and intellectual skills by analysing and discussing key industrial aspects of semiconductor materials applications. On completion of the module, students will be able to: 1. Explain the structure of matter and properties of solids in terms of atomic and molecular bonding. 2. Discuss how the band theory of solids arises when the Schrödinger equation is applied to the behaviour of electrons in solids. 3. Analyse the electrical and magnetic behaviour of solids in terms of the behaviour of their constituent electrons. 4. Analyse the behaviour of semiconductor materials in terms of the properties and behaviour of electrons and holes. 5. Discuss the common semiconductor materials, processing, devices and applications.

This module introduces students to fundamental knowledge in thermodynamics, statistical mechanics and nuclear physics with a focus on transferable skills through problem solving using mathematical modeling. This module also offers the opportunity to analyse nuclear energy power generation and environmental issues in a policy and wider sustainability context, strengthening students’ professional skills and values.

Classical Thermodynamics
Zeroth, first and second laws of thermodynamics, temperature scales, thermal energy and work, internal energy and heat capacity. Classical gas laws. Specific heat, thermal resistance and capacitance and dynamic thermal models of structures such as domestic properties. Thermal cycles, including Carnot, Rankine and Otto cycles; applications to heat engines and heat pumps. Thermodynamic efficiency of heat engines. Entropy, Enthalpy and Helmholtz and Gibbs free energies. Maxwell Relations and Thermodynamic Susceptibilities.

Statistical Mechanics
Kinetic theory of gases; derivation of gas laws, specific heat, and mass, momentum and energy transport coefficients. Statistical mechanical interpretation of entropy. Single and multiple-particle partition function and its relation to thermodynamics. Probability distributions, including Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distribution functions.

Nuclear Energy
The structure of the nucleus, zone of stability and simple nuclear models. Both natural and artificially induced radioactivity, including alpha, beta and gamma radiation. Nuclear fission and nuclear reactors. Nuclear fusion including an introduction to solar nuclear processes and current and future nuclear reactors. Nuclear instrumentation. Nuclear safety.

Nuclear and Thermal Power Sources and their effect on the environment
Thermodynamic efficiency, losses, overall effect on society, Sankey diagrams and simple modelling tools such as the McKay 2050 pathways calculator.

Academic skills when studying away from your home country can differ due to cultural and language differences in teaching and assessment practices. This module is designed to support your transition in the use and practice of technical language and subject specific skills around assessments and teaching provision in your chosen subject. The overall aim of this module is to develop your abilities to read and study effectively for academic purposes; to develop your skills in analysing and using source material in seminars and academic writing and to develop your use and application of language and communications skills to a higher level.

The topics you will cover on the module include:

• Understanding assignment briefs and exam questions.
• Developing academic writing skills, including citation, paraphrasing, and summarising.
• Practising ‘critical reading’ and ‘critical writing’
• Planning and structuring academic assignments (e.g. essays, reports and presentations).
• Avoiding academic misconduct and gaining credit by using academic sources and referencing effectively.
• Listening skills for lectures.
• Speaking in seminar presentations.
• Presenting your ideas
• Giving discipline-related academic presentations, experiencing peer observation, and receiving formative feedback.
• Speed reading techniques.
• Developing self-reflection skills.