Mathematical Physics modules
For a full description, click on the module number or scroll down the page:
- MP460 - Thermodynamics
- MP461 - Statistical Mechanics
- MP463 - Quantum Mechanics 2
- MP464 - Solid State Physics
- MP465 - Electromagnetism
- MP466 - Particle Physics
- MP467 - Astrophysics and Cosmology
- MP468 - Computational Physics 2
- MP469 - Differential Equations and Complex Analysis
- MP471 - Chaos and Nonlinear Dynamics
- MP472 - Quantum Information Processing
Thermodynamics - MP460
Semester 1, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
The aim of this course is to present the key concepts, applications and mathematical tools of thermodynamics. On successful completion of the module, students should be able to:
- analyse problems involving expansion, compression, work and heat in gases and fluids, using the ideal gas law or a more general equation of state
- recall the macroscopic definitions and basic properties of the entropy and the absolute thermodynamic temperature
- state and discuss the first, second an third laws of thermodynamics and apply these laws to problems in thermal physics
- employ the various thermodynamic potentials (enthalpy, energy, Helmholtz and Gibbs free energies) in their appropriate contexts
- derive and use Maxwell's thermodynamic relations
- read phase diagrams of thermodynamic systems and describe the phases (solid, gaseous, liquid) of a simple substance
- analyse phase transitions using Gibbs' phase rule and Clapeyron's equation
Module content
The law of thermodynamics (definition of temperature). Ideal gas law, kinetic theory of gases, Van der Waal's equation. Maxwell-Boltzmann distribution. 1st law of thermodynamics. 2nd law of thermodynamics, (Carnot cycles, entropy, Boltzmann's H-theorem). 3rd law of thermodynamics. Thermodynamic functions (Helmholz and Gibbs functions). Legendre transforms (Gibbs surfaces, Gibbs rule of phases, Gibbs-Duhem relation). Maxwell's equations.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 1 80%. Continuous assessment 20%.
Link to course pages
Staticstical Mechanics - MP461
Semester 2, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
The aim of this course is to present the key ideas, applications and mathematical tools of statistical mechanics. On successful completion of the module, students should be able to:
- Recall the microscopic definitions and basic properties of entropy, temperature and chemical potential
- Give a microscopic interpretation of the second and third laws of thermodynamics
- Deduce and apply the canonical and grand canonical distributions for the states of systems in contact with heat and particle reservoirs
- Calculate partition functions and thermal averages for non-interacting systems such as ideal gases and paramagnets
- Derive the equipartition theorem for the energy and discuss its physical implications and its regime of validity
- Deduce the Fermi-Dirac and Bose-Einstein distributions and apply them in combination with the density of states
Module content
Partition functions (simple sub-systems, ideal gas, simple solids, ferro-magnet, Ising model). Micro-canonical and grand canonical ensembles. Phase transitions and critical phenomena. Bose-Einstein and Fermi-Dirac statistics. Bose-Einstein condensation (superconductors and super-fluids). Information theoretic approach to entropy and partition function.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 2 80%. Continuous assessment 20%.
Link to course pages
Link to lecture notes
Quantum Mechanics 2 - MP463
Semester 1, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
To develop and build on the basis of quantum mechanics learned in the previous module MP363. To master and solve problems with rotational symetry in the two-body bound states of central potentials and to master perturbation theory. On successful completion of the module, students should be able to:
- Define general properties of angular momentum in quantum mechanics, and specifically the relevant observables, their commutation relations, and their eigenvalue equations
- Outline the exact solution of the non-relativistic theory of the hydrogen atom using polynomial method
- Describe the Pauli non-relativistic theory of electron spin
- Perform addition of arbitrary angular momenta using general methods
- Use stationary perturbation theory to calculate first and second order corrections to eigenvalues and eigenvectors
- Describe the spectrum of the hydrogen atom, the relevant degeneracies and quantum numbers, and explain the relativistic origin of the fine and hyperfine structure
Module content
Wave mechanics in one dimension. The one-dimensional harmonic oscillator. Translational and rotational symmetry in the two-body problem. Bound states of central potentials. Perturbation theory
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 2 80%. Continuous assessment 20%.
Link to course pages
Solid State Physics - MP464
Semester 2, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
To use quantum mechanics to master and solve problems in solid state physics. On successful completion of the module, students should be able to:
- Identify crystal structures and associated reciprocal lattices, including Brillouin zone structure
- Describe the criteria for Bragg reflection
- Interpret physical properties of crystals in terms of binding energies and phonons
- Calculate simple transport properties of conductors and semi-conductors
- Reproduce the basic physics underlying band theory
Module content
Crystal Structure; Free Electron Theory of Metals; Energy Bands; Semiconductors; Diamagnetism and Paramagnetism.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 2 80%. Continuous assessment 20%.
Link to course pages
Electrodynamics - MP465
Semester 2, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
To develop and build on the basics of electromagnetic theory learned in previous modules to the point where an understanding of the properties of electromagnetic waves in matter is achieved. To develop relativistic intuition of electromagnetic phenomena. The ability to tackle and solve problems in electrostatics, magnetostatics and electrodynamics. And understanding of how the theory of electromagnetism fits into the special theory of relativity in a 4-dimensional context.
On successful completion of the module, students should be able to:
- Develop multipole expansions due to localised charge distributions in both static and dynamic situations
- Apply Green function techniques to solving electrostatic and radiation problems
- Demonstrate the ability to solve Maxwell's equations in dielectric and magnetic media
- Assess energy transport properties of radiation from simple systems
- Describe the relativistic formulation of electromagnetic theory
Module content
Review of Maxwell's equations in vacuo. Scalar and vector potentials. Multipole expansions in electrostatics. Multipole expansions in magnetostatics. Dielectrics. Diamagnetism and Paramagnetism. Maxwell's equations in the presence of matter. Radiation from simple systems: multipole expansions and energy transport. Relativistic formalism of electromagnetism. Gauge invariance.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 1 80%. Continuous assessment 20%.
Link to course pages
Particle Physics - MP466
Semester 2, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
To introduce students to standard model of particle physics, including the quark model and electro-weak physics. To bring together the theories of quantum mechanics and relativity and show how they are essential to understand the cutting edge of modern physics. A good understanding of the structure of the fundamental building blocks of matter from a modern perspective. An appreciation of how our understanding of the natural world evolves as new experimental data are obtained.
On successful completion of the module, students should be able to:
- Classify particle decays and interactions in terms of the electromagnetic, weak and strong interactions
- Discuss cross sections and resonances
- Apply conservation laws to evaluate the feasibility of specific processes
- Calculate simple cross-sections in elementary particle collisions
- List the elementary particles of the standard model and their quantum numbers
- Reproduce the quark model of hadrons
- Describe CP violation and neutrino oscillation experiments
Module content
Introduction to forces and particles. The four forces; classification of leptons, hadrons, mesons, baryons. Strangeness. The quark model: quark model of mesons and baryons; charm (J/ψ-particle); QCD; asymptotic freedom. Isospin. Neutrinos and neutrino masses. Electro-weak theory and Higgs bosons. Symmetries. Conservation laws. Discrete symmetries (CPT); C, P, T violation. (KL-KS oscillations). Grand unification; Proton decay; Supersymmetry. Current developments:Elementary discussion of string theory, current topics.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 2 80%. Continuous assessment 20%.
Link to course pages
Astrophysics and Cosmology - MP467
Semester 1, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
To give an overview of the structure of the Universe, from the Solar System to galactic super-clusters. To use physical and mathematical tools developed in previous modules to give a good understanding of the basics of stellar structure and Big Bang cosmology. Students will obtain an understanding of the physical phenomena responsible for shaping the Universe. They will appreciate how physical principles can be applied to understand how stars work and how chemical elements are created in stars and in the Big Bang.
On successful completion of the module, students should be able to:
- Summarise the different scales from planets to stars to galaxies to cosmological distances and the basic physics that underpins them
- Explain stellar structure and energy sources in stars
- Describe the basic processes behind star formation and the end points of stellar evolution, including white dwarves, neutron stars and black-holes
- Interpret the evidence for cosmological expansion
- Calculate the dynamics of the Universe at cosmological scales using different equations of state in the Friedmann equations
- Give an overview of the Big Bang theory of cosmological evolution and nucleosynthesis
Module content
Star formation: Jeans mass; PP-chain and CNO cycle. Stellar Structure: hydro-static equilibrium; radiation transport. Eddington limit. Degenerate stars: super-novae; white dwarves; neutron stars; pulsars; black-holes. Binary systems: Gravitational radiation. Galactic structure and evolution: active galactic nuclei; radio galaxies; quasars. Cosmology: the Big Bang; microwave background radiation; nucleo-synthesis; cosmological constant; dark matter; inflation. Astro-particle physics: Cosmic rays; monopoles.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 1 80%. Continuous assessment 20%.
Link to course pages
Computational Physics 2 - MP468
Semester 1, 10 credits, 12 lecture hours, 24 laboratory hours, 16 project work hours
Module objectives
To acquaint the students with methods for solving a wide variety of physical problems using computers. On completing the module, the students shall be able to
- solve ordinary and partial differential equations (initial and boundary value problems) using a variety of methods
- analyse the stability of differentiation schemes for initial-value problems in partial differential equations
- outline the principles behind the most commonly used methods for solving systems of linear equations (Gauss-Jordan elimination, conjugate gradient), their advantages and limitations
- describe and apply the basic principles of Monte Carlo methods
- work in teams to solve a physical problem using computers, and present the results in a coherent, self-contained report.
Module content
Monte Carlo Methods and simulation: random number generators, integration, Metropolis algorithm. Minimisation of multi-dimensional functions: direction set methods, simulated annealing. Linear algebra: Gauss-Jordan elimination, conjugate gradient. Partial Differential Equations. Static solutions/boundary value problems: direct matrix methods, relaxation methods, spectral methods. Time evolution problems: explicit and implicit schemes, stability analysis. Each section will include a range of problems from physics and applied mathematics.
Assessment
Total Marks 100. Two hour written examination at the end of Semester 1 40%; continuous assessment 20%; project 40%.
Link to course pages
Differential Equations and Complex Analysis - MP469
Semester 1, 5 credits, 24 lecture hours, 8 tutorial hours
Module objectives
On successful completion of the module, students should be able to:
- Solve linear inhomogeneous ordinary differential equations, involving elementary and special functions, with initial conditions or boundary conditions
- Recognise and use the basic theory of linear, ordinary differential equations (Sturm-Liouville theory, orthogonal function expansions)
- Solve partial differential equations using separation of variables (with specific applications to the wave equation, the heat equation and Laplace's equation)
- Recognise solutions of the three-dimensional Laplace operator in terms of spherical harmonic functions
- Apply the principles of complex analysis to integrate complex functions using the calculus of residues to compute definite integrals
- Manipulate integral transforms and evaluate Fourier transforms
Module content
Sturm-Liouville theory and expansion in orthogonal bases. Fourier and Laplace transforms. Partial differential equations: Heat equations, Wave equation, Laplace's equation. Spherical harmonics. Green's functions for boundary value problems. Complex analysis up to Cauchy's residue theorem.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 2 80%. Continuous assessment 20%.
Link to course pages
Chaos and nonlinear dynamics - MP471
Semester 2, 5 credits, 24 lecture hours, 12 tutorial hours
Module objectives
The aim of this course is to introduce students to advanced ideas in nonlinear dynamics and chaos. On successful completion of the module, students should be able to:
- formulate and solve problems involving fixed points, limit cycles and strange attractors
- apply basic ideas from statistical mechanics to solve problems in nonlinear dynamics and chaos
- use concepts from thermodynamics and scaling to characterise fractals and multifractals
- apply the mathematical properties of continuous time Brownian motion to solve problems from physical sciences and financial mathematics
Module content
Fixed points, limit cycles and strange attractors. Iterated maps. Quadratic maps: Period doubling, Quasi periodicity, devil's staircase, Farey tree. Universality. Intermittency route to chaos. Hamiltonian systems. Lyapunov exponents, Kolmogorov- Sinai entropy. Concept of Dimension. Fractals and multifractals. Brownian motion.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 2 80%. Continuous assessment 20%.
Quantum information processing - MP472
Semester 2, 5 credits, 24 lecture hours, 12 tutorial hours
Module objectives
To become familiar with fundamental concepts of quantum information processing and its scientific and technological potentials. The motivation and ability to follow research and technological developments in quantum information science and technology and to pursue these topics in advanced study or independent research.
On successful completion of the module, students should be able to:
- Define quantum bits, their composition and elementary quantum operations including measurement
- Use Schmidt decomposition to characterize entangled and separable two-qubit states and their application in teleportation and dense coding
- Demonstrate exponential speedup of quantum computing with the example of the Deutsch-Jozsa quantum computing algorithm
- Compare computational complexity classes of classical, probabilistic, quantum and non-deterministic computing models
- Explain the concept of open quantum systems, their states using density matrix formalism and their operations using operator sum representation
- Present basic single-qubit error processes in the Bloch representation
- Describe quantum error correction on the example of the Shor nine qubit code in the standard and stabilizer formulations
- Explain fault-tolerance criteria and topological quantum computation
Module content
Introduction to classical and quantum information, quantum communication and cryptography, quantum teleportation, physical and conceptual models of computation and computational complexity classes, quantum algorithms, theory of open quantum systems, quantum error correction, fault-tolerant quantum computing, topological quantum computing, physical realization of quantum information processing.
Assessment
Total Marks 100. 1½ hour written examination at the end of Semester 1 80%. Continuous assessment 20%.