Heidelberg 1999 – wissenschaftliches Programm

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 11: Poster

MP 11.1: Poster

Donnerstag, 18. März 1999, 10:00–19:00, MA1

Rate of quantum ergodicity in Euclidean billiards           — •Arnd Bäcker¹, Roman Schubert¹, and Peter Stifter² — ¹Abteilung Theoretische Physik, Universität Ulm, Albert-Einstein-Allee 11, 89069 Ulm — ²Abteilung für Quantenphysik, Universität Ulm, Albert-Einstein-Allee 11, 89069 Ulm

For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdière and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give an explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution.

[1] A. Bäcker, R. Schubert and P. Stifter: Rate of quantum ergodicity in Euclidean billiards, Phys. Rev. E 57 (1998) 5425-5447, erratum ibid. 58 (1998) 5192