- Rate of quantum ergodicity in Euclidean
- A. Bäcker, R. Schubert and P. Stifter
Phys. Rev. E 57 (1998) 5425-5447
Phys. Rev. E 58 (1998) 5192
- 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.
In this work we first give a short introduction to the formulation
of the quantum ergodicity theorem for general observables in terms
of pseudodifferential operators
show that it is equivalent to
the semiclassical eigenfunction hypothesis for the Wigner
function in the case of ergodic systems.
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 a detailed discussion and 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
good agreement with a Gaussian distribution.
- Please refer to the above published version.
(The preprint version of this article can be obtained as
Ulm report ULM-TP/97-8
(September 1997, revised version January 1998))
Last modified: 27 October 2004, 16:21:49
Impressum, © Arnd Bäcker