On subatomic scales, the laws of nature are governed by quantum physics. As one of the main theories of modern physics, it applies to phenomena as diverse as the structure of atoms, light, chemistry, elementary particles, semiconductors and computers, quantum cryptography and teleportation.

This module provides an introduction to the mathematical concepts and foundations of quantum physics. No background in physics will be required.

After a brief revision of relevant parts of classical mechanics, the fundamental Schrödinger equation is derived, and quantum mechanical counterparts of basic quantities such as position, momentum and energy are discussed. The formalism of quantum mechanics is then applied to a number of examples, leading to an understanding of quantum tunneling and the properties of atoms. Further applications to be discussed include Bell's inequalities, entanglement and systems of several particles.

• understand the basic principles of quantum physics, its interpretation and its formulation in terms of operators and Hilbert spaces.

• analyse quantum-mechanical systems by solving the Schrödinger equation in simple cases.

• appreciate the relations between quantum physics and mathematics.

• know and understand the use of symmetry groups in quantum mechanics.

• know and understand the implications of Bell's inequalities.

30 fifty-minute lectures.

During the course, lecture notes will be written and made available to the students, but students are also expected to take notes of lectures.

Students are also expected to undertake at least 120 hours of private study including the solution of exercises.

Skills:

Ability to understand and apply mathematical tools to analyse quantum-mechanical systems.

Transferable Skills:
Ability to recognise, formulate and solve mathematical problems arising from fundamental physics. Ability to use methods from several different areas of mathematics in conjunction.

Formative assessment is carried out in the problem classes. Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.

The summative assessment is the written exam at the end of the term. This gives students the opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

• Review of classical mechanics.

• Fundamentals of quantum mechanics.
  States and observables. Schrödinger equation. Position, momentum, energy. Heisenberg's commutation and uncertainty relations. Commensurable observables and non-commutativity. Hilbert space formulation of quantum mechanics. The interpretation of quantum mechanics.

• Examples of quantum-mechanical systems in one dimension.
  Bound states and the time-independent Schrödinger equation. The spectrum. Scattering and tunneling. The harmonic oscillator.

• Symmetries and quantum mechanics in three dimensions.
  Spherical symmetry, angular momentum. Representations of SO(3) and spherical harmonics. Spin. The hydrogen atom.

• Bell's Inequalities and Entanglement.
  Two-particle systems and tensor products. Bell's inequalities. Entangled states.

• Many-particle systems. (time permitting)
  Fock spaces and many-particle systems. Indistinguishable particles. Second quantisation. Bose/Fermi statistics. The free Bose and Fermi gas.

• Quantum Theory for Mathematicians, Hall, B.C., Springer, 2013

• Quantum Mechanics I, Galindo, A and Pascual, P, Springer, 1989

• Mathematical Foundations of Quantum Mechanics, von Neumann, J., Princeton Landmarks in Mathematics and Physics, 1996

• Methods of Modern Mathematical Physics, Reed, M and Simon, B, Academic Press, 1972