Approach to ergodicity in quantum wave functions

Bruno Eckhardt*, Shmuel Fishman, Jonathan Keating, Oded Agam, Jörg Main, Kirsten Müller

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

90 Scopus citations


According to theorems of Shnirelman and followers, in the semiclassical limit the quantum wave functions of classically ergodic systems tend to the microcanonical density on the energy shell. Here we develop a semiclassical theory that relates the rate of approach to the decay of certain classical fluctuations. For uniformly hyperbolic systems, we find that the variance of the quantum matrix elements is proportional to the variance of the integral of the associated classical operator over trajectory segments of length TH and inversely proportional to TH2, where TH=hρ̄ is the Heisenberg time, ρ̄ being the mean density of states. Since for these systems the classical variance increases linearly with TH, the variance of the matrix elements decays like 1/TH. For nonhyperbolic systems, such as Hamiltonians with a mixed phase space and the stadium billiard, our results predict a slower decay due to sticking in marginally unstable regions. Numerical computations supporting these conclusions are presented for the bakers map and the hydrogen atom in a magnetic field. (c) 1995 The American Physical Society

Original languageAmerican English
Pages (from-to)5893-5903
Number of pages11
JournalPhysical Review E
Issue number6
StatePublished - 1995
Externally publishedYes


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