Synopsis for B7.1a: Quantum Mechanics

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A86) Recommended Prerequisites: Quantum Theory. Calculus of Variations. Classical Mechanics would be useful, but not essential.

Overview

Quantum theory was born out of the attempt to understand the interactions between matter and radiation. It transpired that light waves can behave like streams of particles, but other particles also have wave-like properties. Although there remain deep mathematical and physical questions at the frontiers of the subject, the resulting theory encompasses not just the mechanical but also the electrical and chemical properties of matter. Many of the key components of modern technology such as transistors and lasers were developed using quantum theory, and the theory has stimulated important 20th century advances in pure mathematics in, for example, functional analysis, algebra, and differential geometry. In spite of their revolutionary impact and central importance, the basic mathematical ideas are easily accessible and provide fresh and surprising applications of the mathematical techniques encountered in other branches of mathematics.

This introductory course explores some of the consequences of this, including a treatment of the hydrogen atom.

Learning Outcomes

Students will have gained a sound knowledge of the mathematical ideas related to the development of quantum theory. They will be able to apply mathematical techniques from earlier courses to a range of examples in quantum mechanics.

Synopsis

Maxwell's equations in vacuum and with sources. Lorentz force law. nPlane waves and polarization. Electrostatics and Magnetostatics. Energy density and the Poynting vector. Scalar and vector potentials. Gauge invariance. Maxwell's equations in the Lorentz gauage. The wave equation for potentials.

The mathematical structure of quantum mechanics and the postulates of quantum mechanics. Commutation relations. Poisson's brackets and Dirac's quantization scheme. Heisenberg's uncertainty principle. Creation and annihilation operators for the harmonic oscillator. Measurements and the interpretation of quantum mechanics. Schroedinger's cat. Spin-1/2 particles. Angular momentum in quantum mechanics. Particle in a central potential. The hydrogen atom.

Reading

1. K. C. Hannabuss, Introduction to Quantum Theory (Oxford University Press, 1997). Chapters 1–4, 6–8.

Further Reading

1. A popular non-technical account of the subject:
   A. Hey and P. Walters, The New Quantum Universe (Cambridge, 2003).
2. Also designed for a similar Oxford course:
   I. P. Grant, Classical and Quantum Mechanics, Mathematical Institute Notes (1991).
3. A classical account of the subject which goes well beyond this course:
   L. I. Schiff, Quantum Mechanics (3rd edition, Mc Graw Hill, 1968).
4. Some other books covering similar material:
   B. J. Brandsen and C. J. Joachain, Introduction to Quantum Mechanics (Longman, 1995).
   A. I. M. Rae, Quantum Mechanics (4th edition, Institute of Physics, 1993).