Aims

To introduce the mathematical description of the wave theory of matter and other aspects of basic quantum theory.

To introduce the fundamental concepts and governing equations of classical field theory, with special emphasis on electrodynamics.

Module Summary
Quantum mechanics is the theoretical framework used to describe the most fundamental properties of matter. It has a rich mathematical structure and it has provided the impetus for many advances in mathematics. It also has many practical applications, including the modelling of atoms, molecules and semiconductors. Recently, quantum theory has been used extensively to model superfluids and supercooled gases, and there are even attempts to build computers which function by the laws of quantum mechanics.

This module introduces quantum mechanics in terms of waves and explains how to formulate and solve the Schrodinger equation for matter waves, with appropriate mathematical examples and physical interpretation. Examples include “quantum particles” in different potentials, and their scattering and wave-mechanical interference. It also provides a more formal approach based on simple operator theory, touching on the “measurement problem” and the physics of macroscopic (many-particle) quantum systems.

Classical mechanics of the 18th century has been largely superseded by the ideas of classical field theory. Everything in the physical world, from fundamental particles, to magnetism, light and gravity, is described in terms of a field permeating space and time. The basic ideas of field theory are common to all these applications: moving sources disturb the field, disturbances propagate as waves, and the field reacts back on the sources. The exemplar of field theory is the theory of electric and magnetic fields which forms the core of this module. You will see the power of mathematics in explaining phenomena from electromagnetism and gravity.

Outline Of Syllabus

Wave mechanics overview: mathematical solution of ordinary/partial differential wave equation and wave interference. The collapse of determinism and the uncertainty principle. Schrodinger's equation and concept of quantum-mechanical wavefunction. Mathematical solutions of finite, infinite square wells: energy quantisation, superposition states and the correspondence principle. Wave dynamics on potential barriers. The harmonic oscillator and Hermite polynomials. Formal structure of quantum mechanics: fundamental postulates; operators, eigenvalues and observables; the “quantum measurement” problem; brief introduction to many-particle quantum mechanics and outlook.

Introduction and revision: Scalar and vector fields; Div, grad, curl and the Laplacian;
Conservative and Solenoidal fields-Poincare Lemma;
Laplace’s equation, Poisson’s equation and the Wave equation.
Electrostatics: Coulomb’s Law; Gauss’ Law for discrete charges; Electrostatic potential; Continuous charge distributions.
Magnetostatics: Magnetic forces; Electric currents and Ohm’s law; Biot-Savart Law;
Ampere’s law;
Magnetic vector potential.
Electromagnetism and waves: Lorentz force law; Displacement currents and electromagnetic induction; Maxwell’s equations;
Wave equations;
Plane wave solutions; Permittivity and dispersive waves.
Sources: Multivariable Fourier transforms;
Wave equation with sources;
Retarded potentials;
Simple examples with radiating sources.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.