Date(s) - 28/03/2007
3:30 pm - 4:30 pm
Title: The Wigner Distribution and Classical Chaos
Speaker: Mr. Jamal Sakhr
Institute: Harvard University
Location: ABB 102
The properties of quantum systems whose classical limits are chaotic (either partially or completely) has been a subject of intense interest for more than 30 years, and there are still many open questions. Energy-level statistics, for instance, have been studied extensively but are not yet fundamentally understood. In the formative years, the study of time-reversal invariant quantum systems having strongly chaotic classical limits took precedence. There was a lot of numerical work published between the mid-80s and the mid-90s that showed (among other things) that the probability density of the spacings between adjacent energy levels closely follows the Wigner distribution, which was well known in nuclear physics and in random matrix theory. In view of the cumulative numerical evidence, a Wigner-like energy-level spacing distribution is now widely accepted to be a common property of strongly chaotic Hamiltonian systems (i.e. a ‘quantum signature of classical chaos’).
However, a clear-cut theoretical justification is still lacking. In fact, there has been little material progress (since 1984!) in understanding what aspects or properties of chaos in classical mechanics lead to or imply a Wigner-like energy-level spacing distribution in quantum mechanics. In this talk, I will introduce the idea of doing spatial statistics on the Poincare-Birkhoff surface in classical phase space. For conceptual simplicity, I will consider 2D chaotic infinite well-type potentials (i.e. billiards) for which the Birkhoff surface is simply the boundary of the 3D energy shell. The classical trajectories evolve within the energy shell and under the dynamics repeatedly intersect the boundary (i.e. the Birkhoff surface). It will be shown, by way of example, that the nearest-neighbor spacings between the successive intersection points on the Birkhoff surface obey the Wigner distribution. Thus, it seems that a Wigner spacing distribution is a characteristic statistical property of classical chaos itself! I will also show some other results pertaining to “longer-range” spacing statistics and further interesting correspondences to random matrix theory will emerge.