Physics 599: Independent Study on Quantum Chaos

Time and place: Fridays, 11am-noon, PAS 347.

Feynman's path integral formulation of quantum mechanics provides a natural framework to investigate the connection between the classical dynamics of a system and its corresponding quantum properties. Although chaotic (i.e., nonintegrable) systems cannot be solved exactly in the quantum case, their description in terms of path integrals is in some sense simpler than that of classically integrable systems, since the path integral is dominated by a subset of measure zero of the classical trajectories. This course will provide a thorough introduction to the fascinating field of quantum chaos, which has blossomed in the last three decades. We shall find that revisiting the ``old quantum mechanics'' of Bohr and Sommerfeld, armed with modern path integral techniques and an understanding of classical chaos, is both fun and profitable.

Topics covered:

• Review of basic concepts from quantum mechanics and classical chaos
• Quantization of integrable systems: the role of periodic orbits
• The quantum density of states: some exact trace formulae
• The local density approximation and the Weyl expansion
• Gutzwiller's trace formula for isolated orbits of a chaotic system
• Extensions of Gutzwiller's theory: symmetry and symmetry breaking
• Diverse applications

Student work will consist of periodic homework assignments and an in-class oral presentation on a topic to be agreed upon by the student and the instructor. There will be no exams.

Recommended Textbooks:
Semiclassical Physics, by M. Brack and R. K. Bhaduri, Frontiers in Physics, Vol. 96 (Addison Wesley, 1997).

Chaos in Classical and Quantum Mechanics, by Martin C. Gutzwiller (on reserve in the Science Library).