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# Course units

On this page, you'll find out more about what you can learn in your first, second, third and fourth years at Royal Holloway and the subjects that each of our course units cover.

### PH1110/PH1120 – Mathematics for Scientists

In this module, you’ll cover:

• Basic functions: exponential, logarithm, trigonometric, hyperbolic functions and their inverses.

• Differentiation and integration: limits, methods of differentiation, partial differentiation, Maclaurin and Taylor series, convergence, methods of integration, improper integrals and first and second order differential equations.

• Complex numbers: algebra, geometry of complex numbers, de Moivre’s theorem and functions.

• Determinants and the solution of linear equations.

Vectors: vector algebra and geometry.

Probability: addition and multiplication laws; mean, variance and standard deviation; binomial, Poisson and normal distributions.

• Functions of more than one variable: partial differentiation, differentiation of integrals, extension of Taylor series and multiple integrals.

• Fourier series: full and half range and extension of the interval.

• Coordinate systems.

• Matrices: eigenvaules and eigenvectors

Scalar and vector fields: differentiation of vectors, introduction to grad, div and curl and Gauss’ and Stokes’ theorems.

### PH1320 – Classical Mechanics

Mechanics and Relativity considers one of the most fundamental aspects of Physics rather deeply – how it is that objects move. This first course in mechanics blends the basic physics of the motion of point particles (the principles of conservation of energy and momentum) with the vector calculus mathematics necessary to describe motion properly.

After carefully developing the machinery of Newtonian classical mechanics from first principles, the limits of validity of classical motion are then considered, with short considerations of the interface with quantum mechanics, chaotic motion, statistical physics and other applications.

A longer foray into relativistic mechanics and the consequences of Einstein’s postulates of special relativity then give a deeper understanding of the nature of time. The course continues with a complete deduction of the equations of motion for extended bodies and the principle of the conservation of angular momentum, such that rotation may be included together with the simplifications that come from considering only rigid and symmetric objects. The course emphasises both knowledge of the subject and the problem-solving skills essential in a practising physicist.

• Newtonian mechanics: Motion in 1,2 and 3 dimensions. The constant acceleration equations. Generalised motion. Calculus and vectors as analytical tools. Newton’s laws. Basic concepts (force, mass, work, energy, power, conservation of energy). Conservative forces and the dot product.

• Reference frames: inertial and non-inertial reference frames, the Galilean transformation. Galilean and Einsteinian relativity.

• Special relativistic mechanics: Co-ordinate transformation, Lorentz contraction, time dilation, twin paradox, E = mc2, E2 = c2p2 + m2c4. Collisions, centre of mass, conservation of momentum.

• Rotational motion: The constant angular acceleration equations, Newton’s laws for rotation. Conservation of angular momentum. The cross product. Moments of inertia. Kinetic and potential energy of rotating systems. Gyroscopic precession.

• Conditions for static equilibrium and stability.

• Central force fields: Newton’s Gravitation and Coulomb’s Electrostatic Force Laws, Kepler’s laws, planetary and satellite motion and binary star systems

### PH1620 –Classical Matter

In this course the properties of classical gases, liquids and solids are discussed. A more general aim of the course is to become familiar with concepts, which are relevant in many areas of physics. The description of classical matter will include a macroscopic empirical view as well as more detailed microscopic theories and it will also be shown how macroscopic and microscopic descriptions can be linked. At the start of the course important thermodynamic concepts and macroscopic properties of ideal gases are introduced.

This is followed by the microscopic kinetic theory of ideal gases. The inclusion of collisions between particles allows the describing of transport phenomena and the inclusion of further interactions between particles leads to a theory of real gases. In the area of liquids topics as diverse as surface tension, liquid flow or viscosity will be discussed.

The part on solids has a focus on the topic of elasticity and on thermal effects. Phase transitions between the states of matter are discussed due to their relevance for current research. The description of classical matter serves as an important reference for the description of other areas of physics and the natural sciences in general.

Elementary ideas: Macroscopic descriptions, ideal gases, temperature.Work, internal energy and heat capacity

• Thermodynamic equilibrium and processes, zeroth and first laws of thermodynamics.

• Microscopic models of gases: Kinetic theory, molecular speed distribution, equipartition, transport properties.

• Interatomic forces and properties of solids: forces between atoms, elasticity, expansion, tensile strength, elastic properties of real solids, metals, superconductors.

• Liquids: Properties of fluids, surface tension, liquid flow, viscosity and further properties.

• Changes of state and real gases: Phase equilibrium, van der Waals equation, liquefaction of gases.

• Other states of matter: Disordered systems, polymers, colloids, liquid crystals, plasmas.

### PH1920 – Physics of the Universe

PH1920 gives an introduction to the breakdown of classical physics, which opened the way to quantum mechanics and modern cosmology. The key experiments of the early 20th century that gave rise to quantum theory are discussed and this leads to more advanced discussion of the structure of the atom and introduces quantum mechanics.

The quantum nature of matter is covered, from photons and electrons to the basic concepts of particle physics including quarks, leptons, mesons and baryons. This forms the basis of future courses in particle physics, including the necessary application of special relativity in understanding particle collisions.

### PH1420 – Fields and Waves

• Simple Harmonic Motion: damped, forced resonant harmonic motion. Transients and LCR circuits.

• Coupled oscillators and normal modes.

Wave motion, the wave equation, superposition, transmission, reflection, standing waves and beats, the Doppler Effect, dispersion, phase and group velocity, waves on strings, sound waves: transverse and longitudinal waves and polarisation.

• Coulomb’s Law, Electric fields to the level Ex = -∂V/∂x and grad.

• Parallel plate capacitor formula, reactance.

Magnetic force between two currents.

B-field due to a wire, coil, solenoid etc. Biot-Savart Law and Amperes Law

• Force between two parallel currents, Torque on a coil in a B-field.

D.C and A.C. circuits, complex impedance. Kirchhoff’s laws.

### PH1140/PH1150 – Scientific Skills (lab modules)

• Nine physics experiments/exercises.

• Two formal laboratory reports.

• One review article.

• One talk.

• 72 hours of supervised laboratory time.

• Approximately 78 hours work outside the laboratory on completing analysis and reports.

### PH2130 – Mathematical Methods

• Ordinary differential equations: physical origins, linear equations with constant coefficients, homogeneous and inhomogeneous equations, exact differentials, integrating factors, simple power series method, sines and cosines, Legendre’s equation and Legendre polynomials, Frobenius method, Bessel’s equation and Bessel functions, Sturm-Liouville theorem and orthogonality, orthogonal expansions, Fourier series and other examples.

Partial differential equations: physical origins, the Laplacian, separation of variables, boundary conditions, Cartesian coordinates, polar coordinates, circular membrane, spherical coordinates and Legendre’s equation.

• Integral transforms: Fourier transform and its physical meaning as well as solutions with Fourier transforms.

Gamma function and Dirac delta function.

### PH2150 – Scientific Computing Skills

Basics of numerical calculation on computers: Types of computing - numerical, symbolic, procedural and object-orientated.

Arrays and Matrices: Arrays for storing data, manipulating arrays. Using arrays and matrices (Eigenvector, inversion).

Plotting and visualization: Scatter plots, Histograms and Multi-dimensional plots. Data analysis: Mean/mode/median; Variance, RMS, kertosis, moments; Fitting (least squares); Regression.

• Advanced data analysis: Fourier series, Fourier transform and Smoothing.

• Programming: Control structures (if, for, while, return etc). Functions and program layout

• Simulation: Evaluation of simple and complex expressions. Monte Carlo. Numerical integration.

• Differential equations: Difference methods. Leap-frog method and Runge Kutta.

Monte Carlo methods:Metropolis algorithm. Integrals, integral estimation.

• Linear equations: Solutions, eigenvalues. Diagonalisation and Factorisation.

Project report: example illustrating physics from 2nd year core.

### PH2210 – Quantum Mechanics

Electron diffraction experiments show that the electron, which is classically thought of as a particle, can be equally well described as a wave. In this course we will show that any small particle can be described by wavefunction that obeys Schrödinger’s equation, a differential equation that like Newton's equation (F=ma) is important in determining in what happens next - that is how the wavefunction evolves with time.

Having defined the wavefunction the main purpose of the course is to understand how the electrons in an atom come to be in particular orbits, which determines the atom's chemical and physical properties. This requires an understanding of angular momentum and spin, the latter being a quantum mechanical concept that has not classical analogy. Quantum mechanics is probably the most successful theoretical description of the world as we know it, and its development in the early 20th century involved physics greats such as Schrödinger, Born, Dirac, Bohr, Heisenberg and Pauli.

• Review of the failure of classical physics and the introduction of quantum ideas; the uncertainty principle. Schrodinger’s equation; the wave function and its interpretation.

Operators, eigenvalues, eigenfunctions, expectation values; commutation relations and non-commutation.

Applications of Schrödinger’s equation in one-dimension: free particle, square well, harmonic oscillator; potential step, barrier and tunnelling (including alpha decay).

• Applications of Schrödinger’s equation in three dimensions: particles in a rectilinear box, angular momentum, the hydrogen atom.

The exclusion principle, atomic structure and the periodic table. Mixed states. Simple perturbation theory.

### PH2310 - Optics

Definition of Fourier transforms and their properties; Convolution and Transfer functions; Various applications of Fourier Transforms and Solutions of Differential Equations using Fourier Transforms.

• Geometrical optics. Fraunhofer and Fresnel diffraction. Optical instruments: image formation and quality. Rayleigh criterion and resolving power, aberrations. Interference. Two-beam and multiple-beam effects. Interferometers. Thin films. Dispersion in dielectric media. Normal and anomalous dispersion. Scattering of e-m radiation. Polarisation by scattering and other means.

• Masers and lasers: history; spontaneous and stimulated emission; Einstein coefficients; practical details of lasers and laser systems; properties of laser light. Fourier optics and diffraction; Huygens-Fresnel theory of diffraction; Gaussian beams; spatial and temporal coherence; applications to optical imaging, image processing and Fourier Transform spectroscopy.

### PH2420 - Electromagnetism

The electromagnetic force is one of the four fundamental forces of nature, and as such pervades all of physics. All of electromagnetism is embodied in Maxwell’s equations, and an understanding of these equations and their consequences is the objective of this course.

Studies in electricity and magnetism led ultimately to the prediction and finally the generation and detection of electromagnetic radiation. All this and more is described in this course.

• Vector calculus treatment of Electrostatics: the electric field, Coulomb’s and Gauss’ laws, electric field energy, equations of Poisson and Laplace.

• Steady currents: continuity equation, Kirchhoff’s laws, Laplace’s equation in conductors.

• Vector calculus treatment of the magnetic effects of currents:  Biot-Savart law, magnetic field, Ampère’s law, Lorentz force and energy of a magnetic field.

Induction: Faraday’s law, Lenz’s law and Displacement current.

Electric and magnetic dipoles and the electromagnetism of matter. Macroscopic fields E, D, B and H and their boundary conditions at interfaces.

• Maxwell’s equations; plane waves in free space and in media; reflection, refraction, polarisation. Poynting vector. The EM spectrum and Radiation pressure.

• Propagation in lossy media: dielectric loss, ohmic loss, skin depth and electromagnetic screening.

• Electromagnetic potentials. Gauge freedom. Scalar and vector potentials for stationary and time varying problems. Radiation from dipoles.

• Incompatibility of Maxwell’s equations and Newton’s laws. Resolution by Lorentz transformation and special relativity.

### PH2510 – Atomic and Nuclear Physics

• Single-electron atoms.

• Multi-electron atoms.

• Atomic spectra.

• X-ray spectra.

• Hyperfine structure.

• Experimental techniques.

• Nuclear models.

• Radioactive decay: alpha, beta and gamma decay.

• Nuclear reactions: Fission, fusion, chain reaction.

• Interaction of radiation with matter.

### PH2520 – Particle Detectors and Acclerators

• Detectors: Interaction of particles in matter, energy loss processes. General characteristics of detectors: sensitivity, resolution, efficiency, dead time. Principles of operation of ionisation, scintillation, semiconductor and Cherenkov detectors. Applications to the tracking, energy measurement and identification of particles in particle physics experiments. Study of all multi-purpose detectors in high energy physics, integrating several of the detection methods above.

Accelerators: Introduction to the basic physical principles underlying the operation of modern accelerators. Transverse motion of particles in a beam; beam optics: bending and focusing elements. Accelerating cavities. Synchrotron radiation. Circular and linear accelerator designs. Methods for measurement of energy, luminosity and polarisation in colliders. Study of the design and the operation of e+e- and hadron colliders. Introduction to some applications in the area of medical physics.

### PH2610 – Classical and Statistical Thermodynamics

The first section of the course completes the classical treatment of thermal physics started in the first year course, Classical Matter. The power of this treatment is its generality and the fact that it does not rely on microscopic models. In the second section of the course, a microscopic understanding of thermal physics is developed, building on the kinetic theory treatment of a classical gas introduced in Classical Matter, by the introduction of elementary quantum mechanics.

Familiar results will be obtained, for example the thermodynamic properties of an ideal gas will be derived from the solutions of Schrödinger’s equation for particles in a box, and exotic new phenomena are encountered, such as negative temperature, superfluidity and superconductivity. This treatment links in an elegant manner the quantum world with everyday observables of systems containing large numbers of particles.

• Thermodynamic equilibrium and processes. The zeroth and first law of thermodynamics.

• Temperature, heat and work. Properties of ideal gases. Counting microstates, order and entropy. The second law of thermodynamics. The statistical mechanics of localised systems: the spin ½ paramagnet, Einstein model of a solid. Boltzmann distribution, partition function.

• Entropy in thermo-dynamics. Cyclic processes and heat engines. Thermodynamic potentials: equilibrium, Maxwell’s relations and their applications. Statistical mechanics of gases, density of states. Calculation of properties of gases in the classical limit, partition function, entropy, equation of state, Maxwell-Boltzmann velocity distribution. Collisions and transport properties. Fermi gases: the Fermi Dirac distribution function. Application to electrons in metals and liquid 3He. Bose gases: The Bose-Einstein and Planck distribution functions.

• Applications to photons (black body radiation), phonons (Debye model), liquid 4He. Phase equilibrium. Generic phase diagram. Conditions for phase coexistence. Clausius-Clapeyron equation. Applications to real systems, including 3He. The third law of thermodynamics and the unattainability of absolute zero.

### PH2710 – The Solid State

• Structure, symmetry and properties of solids.

• Mechanical properties: plastic deformation and dislocations.

• Methods of probing solids. X-ray diffraction.

• Thermal properties of solids and  phonons.

• Electrical properties of metals, alloys and semiconductors.

• Quantum theory of solids. Energy bands. Bloch theorem.

• Fermiology.

• Intrinsic and extrinsic semiconductors. p-n junctions. Hall effect.

• Magnetic properties: dia-, para-, ferro- and anti-ferro-magnetism.

### PH2900 - Astronomy

• Observing the Universe: Observations in different wavelengths.

• Earth and space-based telescopes (and related instruments) and their merits and limitations.

• Basic parameters and measurements in Astronomy: coordinate systems, timekeeping systems, magnitude, angular size, luminosity, colour, mass, distance, temperature, standard candles and redshift.

• The structure and dynamics of the solar system.

• The contents of the solar system: the planets and their moons, rings, asteroids, comets, dust and the solar wind.

• An introduction to planetary geology.

### PH3110 – Experimental or Theoretical Project

This course should provide the high point of the three year physics degree. It gives students the opportunity to use their scientific knowledge, their ability to plan and execute an extended experimental or theoretical investigation, and use all their communication skills to describe their results.

Working together with a member of the academic staff students carry out extended experimental or theoretical work, in physics, electronics or astrophysics. They present their work via an oral presentation and write a report, which they can then show at career interviews and discuss its contents with confidence.

### PH3520 – Particle Physics

This course provides an overview of the physics of elementary particles, with particular focus on the current Standard Model of electroweak and strong interactions. The course includes a brief overview of experimental methods (particle detectors and accelerators) and reviews the primary discoveries that have led to our current understanding of quarks, leptons, gauge bosons, and their interactions.

Feynman diagram techniques are used to calculate cross sections and decay rates for observable processes, and these predictions are compared with experimental results. A brief description of the predicted properties of the Higgs boson is given and ongoing searches for this particle are described.

### PH3710 – Semiconductors and Superconductor

This course deals with the properties of two important classes of solids: semiconductors and superconductors. Their physical description requires advanced quantum mechanical and many-particle concepts. The semiconductor module uses basic solid-state theory to understand their properties, and understand how these properties can be tuned by varying the bulk doping of the semiconductor. Semiconductors can be doped either p-type or n-type, allowing the simplest semiconductor device, a pn junction to be fabricated.

As well as looking at the optical and transport properties of semiconductors, the principles of how real devices are fabricated are discussed. In the module on superconductors their important properties like perfect conductivity or the expulsion of the magnetic field are discussed using the phenomenological approach of Ginzburg-Landau theory. The theory includes a complex order parameter, whose phase is used to describe interference phenomena in Josephson junctions and SQUIDS, which allow the detection of tiny magnetic fields. Current research on unconventional superconductors with high transition temperatures will be summarised.

### PH3730 – Modern topics of Condensed Matter

Three areas of condensed matter physics are discussed: probes of condensed matter, magnetism, and soft matter. Starting from basic concepts each topic will be developed to allow a connection with current research. Students will write an essay on a specific research topic. The module on probes of condensed matter focuses on the high spatial resolution methods for probing structure, composition, surface properties and electronic structure of solids including Scanning Electron Microscopy, X-ray spectroscopy, Scanning Tunnelling Microscopy, Atomic Force and Magnetic Force Microscopy and Scanning Near-field Optical Microscopy. Major interactions of electron beams and other scanning probes with condensed matter will be discussed.

The magnetism module contains a description of four different types of magnetic systems: materials with localised and delocalised magnetic moments, which can either be non-interacting or interacting. This includes a description of magnetic insulators and metals. The relevance of magnetic interactions for giant magneto resistance or superconductivity with its many current or potential applications in information technology and energy transport will be discussed. The soft matter module concerns the interactions between colloidal particles, their phase diagrams, self-diffusion and rheological behaviour. The theory of polymer chains is also covered. This includes coarse-graining, polymer shape measures, dynamics and viscosity as a function of chain length, including reptation. Reference to recent advances and practical applications will be made.

### PH3930 – Particle Astrophysics

PH3930 gives an introduction to the modern field of particle astrophysics. Building on the particle physics course PH3520, the use of Feynman diagrams to understand decay rates and cross sections is described and key processes are explored, which are instrumental in the development of the early universe. Supersymmetry and Grand Unified Theories are described, together with their implications for high-energy experiments at the LHC and also for dark matter and high-energy neutrino events from space.

Terrestrial neutrino experiments are currently a hot topic and the principle behind them is covered, together with a selection of the latest measurements. Searches for dark matter are also a very active research field and the latest ideas are covered and implications from cosmological measurements are also discussed. The development of the early universe starting from the big bang is derived, starting from the Friedman equation and employing the standard model of particle physics. Throughout, the links between cosmology and particle physics are stressed and the latest measurements from the field are presented.

### PH4100 - Major Project

This is a major individual project making up a quarter of the workload of final year MSci students. Projects are associated with the research of the department and can cover experimental, theoretical or computational physics.

### PH4110 - Research Review

You choose a topic related to problems of current research interests in physics. You will consult original papers and write a report, complete with references, which is expected to demonstrate a high level of understanding.

PH4201 - Mathematical Methods for Theoretical Physics **

PH4205 - Lie Groups and Lie Algebras **

PH4211 - Statistical Mechanics

This course is a review of equilibrium statistical mechanics, covering elements such as the grand canonical ensemble, Bose and Fermi distribution functions, classical partition functions and Weakly Interacting Systems, to name just a few.

PH4226 - Advanced Quantum Theory **

PH4317 - Galaxy and Cluster Dynamics **

PH4421 - Atom and Photon Physics **

PH4427 - Quantum Computation & Communication **

PH4431 - Molecular Physics **

PH4442 - Particle Physics **

### PH4450 - Particle Accelerator Physics

This course aims to introduce you to the key concepts of modern particle accelerators by covering topics including:

• Introduction: history of accelerators, basic principles including centre of mass

• Characteristics of modern colliders; LEP, LHC, b-factories

• Transverse motion, principles of beam cooling

• Strong focusing, simple lattices

• Longitudinal dynamics

• Multipoles, non-linearities and resonances

• Radio Frequency cavities, superconductivity in accelerators

• Applications of accelerators; light sources, medical uses

• Future: ILC, neutrino factories, muon collider, laser plasma acceleration.

PH4472 - Order & Excitations in Condensed Matter **

PH4473 -Theoretical Treatments of Nano-Systems **

PH4474 - Physics at the Nanoscale **

PH4478 - Superfluids, Condensates & Superconductors

PH4500 - Standard Model Physics and Beyond **

### PH4512 - Nuclear Magnetic Resonance

This introduces you to the principles and methods of nuclear magnetic resonance by looking at:

• Introduction: static and dynamic aspects of magnetism, Larmor precession, relaxation to equilibrium, T1 and T2, Bloch equations.

• Pulse and continuous wave methods: time and frequency domains. Manipulation and observation of magnetisation, 90º and 180º pulses, free induction decay.

• Experimental methods of pulse and CW NMR: the spectrometer, magnet. Detection of NMR using SQUIDs.

• Theory of relaxation: transverse relaxation of stationary spins, the effect of motion. Spin lattice relaxation.

• Spin echoes: ‘violation’ of the Second Law of Thermodynamics, recovery of lost magnetisation. Application to the measurement of T2 and diffusion.

• Analytical NMR: chemical shifts, metals, NQR.

• NMR imaging: Imaging methods. Fourier reconstruction techniques. Gradient echoes. Imaging other parameters.

### PH4515 - Computing & Statistical Data Analysis

You'll be introduced to programming language using C++. This course covers:

• Introduction to C++ and the Unix operating system.

• Variables, types and expressions.

• Functions and the basics of procedural programming.

• I/O and files.

• Basic control structures: branches and loops.

• Arrays, strings, pointers.

• Basic concepts of object oriented programming.

• Probability: definition and interpretation, random variables, probability density functions, expectation values, transformation of variables, error propagation, examples of probability functions.

• The Monte Carlo method: random number generators, transformation method, acceptance-rejection method.

• Statistical tests: significance and power, choice of critical region, goodness-offit.

• Parameter estimation: samples, estimators, bias, method of maximum likelihood, method of least squares, interval estimation, setting limits, unfolding.

PH4534 - String Theory and Branes **

PH4541 - Supersymmetry & Gauge Symmetry **

PH4600 - Stellar Structure & Evolution **

PH4602 - Relativity & Gravitation **

PH4603 - Astrophysical Fluid Dynamics **

PH4630 - Planetary Atmospheres **

PH4640 - Solar Physics **

PH4650 - Solar System **

PH4660 - The Galaxy **

PH4670 - Astrophysical Plasmas **

PH4680 - Space Plasma & Magnetospheric Physics **

PH4690 - Extrasolar Planets & Astrophysical Discs **

PH4800 - Molecular Biophysics **

** taught at either King's, Queen Mary or University College.