Mathematical Physics - U6UMTHPY

This module will introduce you to the mathematical language behind the classical mechanics describing our universe. You will learn about Lagrangians and Hamiltonians, the starting place from which we can determine the dynamics of complicated systems, like pendula and planets orbiting the sun, as well as the origin of conserved quantities such as energy and momentum.

This is a fun module. At school you learnt Kepler’s Laws, Newton’s Law of Gravity, and F=ma, but how can you derive these amazing results? Where do they come from?

Here you will find out, as we introduce you to the mathematical language behind the classical mechanics describing our universe. You will learn about Lagrangians and Hamiltonians, the starting place from which we can determine the dynamics of complicated systems, like pendula and planets orbiting the sun, as well as the origin of conserved quantities such as energy and momentum. For two hours a week we will take you into the mathematics and ideas of giants like Newton, Euler, Lagrange, Noether and Hamilton.

Among many exciting things, you will study:

• Newton’s Laws and deriving the orbits predicted by Kepler
• Lagrangians and Hamiltonians, the building blocks behind classical mechanics
• The Euler-Lagrange equations describing the dynamics behind classical systems
• Rigid bodies – introducing moments of inertia, centre of mass and more so that we can apply these results to many particle rigid systems, like pendulums and even you
• Constraints – how to determine the dynamics of a system where it is constrained, for example, the motion of an explorer constrained to be on the surface of the earth
• The motion of charged particles, like electrons in an electromagnetic field
• Hamilton’s equations as an alternative way to determine the dynamics of a system, particularly useful when we are searching for conserved quantities like angular momentum
• Spinning tops – what? You heard right, the vital roles of gyroscopes in our life are understood by 5-year-olds, but the mathematics certainly is not. Thanks to this course, now you can understand that as well.

This module introduces basic techniques in numerical methods and numerical analysis which can be used to generate approximate solutions to problems that may not be amenable to analysis. Specific topics include:

• Implementing algorithms in Matlab
• Discussion of errors (including rounding errors)
• Iterative methods for nonlinear equations (simple iteration, bisection, Newton, convergence)
• Gaussian elimination, matrix factorisation, and pivoting
• Iterative methods for linear systems, matrix norms, convergence, Jacobi, Gauss-Siedel
• Interpolation (Lagrange polynomials, orthogonal polynomials, splines)
• Numerical differentiation & integration (Difference formulae, Richardson extrapolation, simple and composite quadrature rules)
• Introduction to numerical ODEs (Euler and Runge-Kutta methods, consistency, stability) 

In this module you will learn how Newtonian mechanics can be developed into the more powerful formulations due to Lagrange and Hamilton and be introduced to the basic structure of quantum mechanics. The module provides the foundation for a wide range of more advanced modules in mathematical physics.

In this module you will learn how the same physics that works on Earth – gravity, electromagnetism, thermodynamics, optics, quantum physics, atomic and nuclear physics – is used to understand stars. You will explore the most important physical processes occurring in stars of different types. You will then use this knowledge to build mathematical models of stars and to understand their internal structure, their formation, evolution, and death.

You’ll study:

• How astronomers measure the most important properties of stars such as their mass, size, distance, brightness, temperature, chemical composition and age. This module will then teach you how physics is able to explain these properties.
• How energy is generated inside stars through nuclear fusion, and how it is transported to the surface to make stars shine.
• How to write the equations that describe the structure of stars, and how to use them to build mathematical models that explain their properties and evolution.
• How stars are born, how they evolve with time, how long they live, how they die, and what remnants they leave behind. You will be able to understand, for instance, how supernovae explode and how some black holes form.

In the first semester of this module, you'll participate in a series of lectures, workshops, and practical classes. Some sessions will build on the "Discovery" strand from Investigations in Physics I and II, while others will provide training on various computational techniques relevant to the offered projects. This foundational training will prepare you for the hands-on work to come.

During the second semester, you'll work in pairs on a computational project chosen from a published list. By the end of the module, you'll have successfully used advanced computational techniques and developed a broader knowledge of computational physics, including recent developments from current literature. You'll also enhance your independent learning skills and further develop your investigative abilities by designing and conducting a computational study, analysing its results, and drawing critical conclusions. This module will prepare you for a career in computational physics, equipping you with practical experience and a deep understanding of the field.

In this module, you'll explore how physical behaviour at the nanoscale can differ radically from macroscopic systems, drawing on principles from classical and quantum mechanics, electromagnetism, and thermodynamics. You'll learn about techniques such as scanning probe microscopy and optical trapping, which are used to measure and manipulate nanoscale objects. The module will also introduce you to soft condensed matter systems, including amorphous solids and complex fluids.

Additionally, you'll discover how nanoscale physics contributes to our understanding of biomolecular systems. By the end of the module, you'll have a comprehensive grasp of the unique properties and behaviours of nanoscale materials, and the advanced techniques used to study them. This module will equip you with the knowledge and skills to apply nanoscale physics in various scientific and technological contexts.

This module builds on your knowledge of quantum theory, with a focus on how quantum systems evolve over time. It develops the mathematical formalism of quantum mechanics and introduces key physical models and applications, including quantum computing.

The module begins with a review of Dirac notation and its use in formulating quantum problems. It then covers the dynamics of operators and wavefunctions, along with approximate methods for solving time-dependent problems.

Finally, you will be introduced to open quantum systems, quantum systems that interact with their environment, and the mathematical tools used to study them.

This module begins by exploring the techniques for magnetic resonance imaging (MRI) and spectroscopy (MRS). In the first semester, it covers the classical description of MRI, focusing on how images with contrast are generated to assess structure and function in the body for clinical diagnoses.

In the second semester, the module delves deeper into the physics of nuclear magnetic resonance (NMR) and MRI, presenting a quantum description of the technique and examining the clinical applications of MRS. It then shifts to other imaging methods commonly used in hospitals, including ultrasound and optical coherence tomography (OCT). For each technique, you will learn about the transmitters, detectors and safety considerations involved.

Cosmology is the scientific study of the Universe as a whole. It aims to understand what the Universe is made of, and its evolution from the Big Bang until today (and into the future).

You’ll study:

• observational evidence for the Big Bang
• how the expansion of the Universe depends on its contents and geometry
• how the contents of the Universe evolve as it expands and cools
• dark matter and dark energy: observational evidence and the latest theoretical models
• inflation, a proposed period of accelerated expansion in the very early Universe

This module introduces spacetime and gravity, as explained by Einstein's theories of relativity.

You will learn that when velocities approach a significant fraction of the speed of light, spatial distance and elapsed time become relative to the observer. The relativistic laws of mechanics are presented within a unified framework of four-dimensional spacetime, and key implications, such as Einstein's famous equation E=mc2E=mc2, are explored. 

The module also covers the concept of gravitational effects, which require spacetime to be warped or curved. You will study the relevant mathematics to describe this curvature and its physical effects, using the Schwarzschild solution and the corresponding black hole as a key example.

Mathematics can be usefully applied to a wide range of applications in medicine and biology.

Without assuming any prior biological knowledge, this module describes how mathematics helps us understand topics such as:

• population dynamics
• biological oscillations
• pattern formation
• nonlinear growth phenomena

There is considerable emphasis on model building and development.

This course builds on the foundations of quantum mechanics introduced in the module MATH2013. It further develops the fundamental theory so that it applies to more general problems, such as those involving spin, and introduces key calculational approaches, such as those underlying angular momentum, the hydrogen atom, scattering problems and approximation methods such as perturbation theory.

20 credits in the Autumn Semester

In this module a variety of techniques of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming.

These techniques have a wide range of applications to real world problems, in which a process or system needs to be made to perform optimally.

This module explores the physical processes involved in the most extreme environments known in the Universe. Among the objects studied are neutron stars, black holes, supernova explosions, and active galactic nuclei.

This module explores how electrical currents enable communication within the body. You will learn the biological processes in nerve cells that allow electrical signalling, as well as how the ear converts sound and the eye converts light into electrical signals. The module also covers how the heart uses electrical activity to pump blood and how muscles function.

Building on this, you will study how electrical signals from the heart, muscles, and brain can be measured and interpreted through surface recordings. The module also introduces magnetoencephalography (MEG), explaining how tiny magnetic fields generated by electrical currents in the brain can be detected, and the physical design of sensors and specialised rooms required for these measurements.

This module covers the Standard Model of particle physics, detailing the known particles of nature and the theoretical concepts behind them. It introduces natural units, relativistic field equations, and the mathematics of group theory and Lie groups.

Building on this, you will explore the Standard Model's particle content and their interactions through Feynman diagrams. The module will also cover gauge symmetries, applied to the force-carrying particles, as well as discrete symmetries, quantum mixing, and the Higgs mechanism.

Macroscopic matter displays a wide range of phases, from the familiar solids, liquids, and gases to more exotic phases like liquid crystals, superfluid and superconductors. 

In this module, you will learn how these ordered phases are described through broken symmetries, which helps in understanding their properties and phase transitions. You will study various mean-field approaches used to analyse these phases. Finally, you will explore how moving beyond the mean-field approach reveals the emergence of universal critical behaviour.

This module introduces you to the physics and applications of Semiconductors. Semiconductors are key materials of the current Information Age. They enabled most of the devices and technologies we use everyday, such as computers, internet, mobile phones. Semiconductors help us to mitigate global warming, data theft, end of the Moore’s law and other global challenges.

This module includes detailed overview of the Semiconductors past, present and future, and provides skills and knowledge essential for a future Semiconductor researcher or engineer.

You’ll study:

• Physics and applications of conventional semiconductor materials and devices, for example p-n diodes and field-effect transistors
• Physics and applications of novel semiconductor materials, quantum materials, nanostructures, low dimensional materials, such as graphene and quantum dots
• Current and future semiconductor challenges and technologies, such as efficient solar cells, ultrasensitive phone cameras and quantum computers.

The course introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics. The course explores how classical concepts enter quantum mechanics, and how they can be used to find approximate semiclassical solutions.

In classical dynamics we discuss full integrability and basic notions of chaos in the framework of Hamiltonian systems, together with advanced methods like canonical transformations, generating functions and Hamiltonian-Jacobi theory.

In quantum mechanics we recall Schrödinger’s equation and introduce the semiclassical approximation. We derive the Bohr-Sommerfeld quantization conditions based on a WKB-approach to the eigenstates. We will discuss some quantum signatures of classical chaos and relate them to predictions of random-matrix theory.

We will also introduce Gaussian states and coherent states and discuss their semiclassical dynamics and how it is related to the corresponding classical dynamics.  An elementary introduction to complete descriptions of quantum mechanics in terms of functions on the classical phase space will be given.