Synopsis for C7.1b: Quantum Theory and Quantum Computers

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit, OSS paper code 2A78

Prerequisites

B7.1a Quantum Mechanics.

Overview

This course builds directly on the first course in quantum mechanics and covers a series of important topics, particularly features of systems containing several particles. The behaviour of identical particles in quantum theory is more subtle than in classical mechanics, and an understanding of these features allows one to understand the periodic table of elements and the rigidity of matter. It also introduces a new property of entanglement linking particles which can be quite widely dispersed.

There are rarely neat solutions to problems involving several particles, so usually one needs some approximation methods. In very complicated systems, such as the molecules of gas in a container, quantum mechanical uncertainty is compounded by ignorance about other details of the system and requires tools of quantum statistical mechanics.

Two state quantum systems enable one to encode binary information in a new way which permits superpositions. This leads to a quantum theory of information processing, and by exploiting entanglement to other ideas such as quantum teleportation.

Learning Outcomes

Students will be able to demonstrate knowledge and understanding of quantum mechanics of many particle systems, statistics, entanglement, and applications to quantum computing.

Synopsis

Identical particles, symmetric and anti-symmetric states, Fermi-Dirac and Bose-Einstein statistics and atomic structure.

Heisenberg representation, interaction representation, time dependent perturbation theory and Feynman–Dyson expansion. Approximation methods, Rayleigh-Schr{ö)dinger time-independent perturbation theory and variation principles. The virial theorem. Helium.

Mixed states, density operators. The example of spin systems. Purification. Gibbs states and the KMS condition.

Entanglement. The EPR paradox, Bell's inequalities, Aspect's experiment.

Quantum information processing, qubits and quantum computing. The no-cloning theorem, quantum teleportation. Quantum logic gates. Quantum operations. The quantum Fourier transform.}

• K. Hannabuss, Introduction to Quantum Mechanics (oup, 1997). Chapters 10–12 and 14, 16, supplemented by lecture notes on quantum computers on the web.

Further Reading

• A popular non-technical account of the subject: A. Hey and P. Walters, The New Quantum Universe (Cambridge, 2003).
• Also designed for an Oxford course, though only covering some material: I. .P Grant, Classical and Quantum Mechanics, Mathematical Institute Notes (1991).
• A concise account of quantum information theory: S. Stenholm and K.-A. Suominen, Quantum Approach to Informatics (Wiley, 2005).
• An encyclopaedic account of quantum computing: M. A. Nielsen and I. L. Chuang, Quantum Computation (Cambridge University Press, 2000).
• Even more paradoxes can be found in: Y. Aharanov and D. Rohrlich, Quantum Paradoxes (Wiley–VCH, 2005).
• Those who read German can find further material on entanglement in: J. Audretsch, Verschränkte Systeme (Wiley–VCH, 2005).
• Other accounts of the first part of the course: L. I. Schiff, Quantum Mechanics (3rd edition, Mc Graw Hill, 1968). B. J. Bransden and C. J. Joachain, Introduction to Quantum Mechanics (Longman, 1995). A. I. M. Rae, Quantum Mechanics (4th edition, Institute of Physics, 1993). John Preskill's on-line lecture notes (\href{http://www.theory.caltech.edu/~preskill/ph219/index.html}{http://www.theory.caltech.edu/~preskill/ph219/index.html}).