Module manager: TBC
Email: TB
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
None
| MATH1000 | Core Mathematics |
| MATH2350 | Vector Calculus and Partial Differential Equations |
MATH3424 MATH3385
This module is not approved as a discovery module
The material world is composed of countless microscopic particles. When three or more particles interact, their dynamics is chaotic, and impossible to predict in detail. Further, at the microscopic atomic-scale particles behave like waves, with dynamics known only statistically. So, why is it that the materials around us behave in predictable and regular ways? One reason is that random behaviour on the microscopic scale gives rise to collective behaviour that can be predicted with practical certainty, guided by the principle that the total disorder (or entropy) of the universe never decreases. A second reason is that the mathematics of quantum mechanics provides incredibly accurate predictions at the atomic scale. This module studies calculations involving both entropy and quantum mechanics, as applied to the matter that makes up our world.
This module will introduce students to the mathematical tools for modelling and understanding the relationship between microscopic and macroscopic properties of the natural world. Students will gain an understanding of how the macroscopic properties of matter emerge from the microscopic processes within it. They will learn how to model the quantum uncertainty of microscopic particles and will appreciate the need for a statistical approach to the dynamics of interacting systems involving large numbers of degrees of freedom, which ultimately determines the "arrow of time" via the direction of physical processes.
On successful completion of the module students will have demonstrated the following learning outcomes relevant to the subject: 1. define and evaluate entropies; 2. know and use the statistical definition of temperature; 3. calculate expectation values in microscopic models using the Boltzmann distribution; 4. find the partition function for a number of idealized models; 5. understand the basic principles of quantum mechanics and be able to apply them in simple physical situations; 6. calculate the quantum mechanical wave function and probability distribution in a given state; 7. calculate quantum mechanical expectation values of observables; 8. predict how the state of an undisturbed quantum system evolves with time.
1. Brief survey of Newtonian mechanics; reversibility, irreversibility, chaos, and the arrow of time; 2. Definitions of entropy, statistical weight and extensivity; 3. Energy, entropy, and temperature; 4. The Boltzmann distribution and the partition function; 5. The need for quantum mechanics and the wave-particle duality; 6. The Schrödinger equation, wavefunctions and probabilities; 7. Eigenstates and quantised energy levels for simple 1-d potentials; 8. Quantum tunnelling and barrier penetration. 9. The harmonic oscillator 10. Bosons, fermions, and quantum statistics 11. Applications to physical systems Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar: 1. Classical theory of gases; 2. Diffusion in a potential, equilibrium distribution and barrier crossing; 3. Lasers; 4. Engines and heat pumps 5. Polymer chains and lattice models; 6. Semiconductors, superconductors and superfluids.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 44 | 1 | 44 |
| Private study hours | 156 | ||
| Total Contact hours | 44 | ||
| Total hours (100hr per 10 credits) | 200 | ||
4-6 examples sheets throughout semester 2. Opportunity to submit solutions (~10 pages) for marking and feedback. Solutions presented in lectures afterwards
Check the module area in Minerva for your reading list
Last updated: 30/04/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team