[工程科技]Autocorrelation function of eigenstates in chaotic and mixed systems.pdf

ULM–TP/01–04 HPL–BRIMS–2001–11 April 2001 Autocorrelation function of eigenstates in chaotic and mixed systems 1 0 0 2 n u J Arnd B¨acker1,2 and Roman Schubert3 3 1 ] D C 1) School of Mathematics, University of Bristol n. University Walk, Bristol BS8 1TW, UK i l n 2) BRIMS, Hewlett-Packard Laboratories [ Filton Road, Bristol BS12 6QZ, UK 1 v 3) Abteilung Theoretische Physik, Universit¨at Ulm, 8 1 Albert-Einstein-Allee 11, D–89069 Ulm, Germany 0 6 0 1 0 / n i l n : v Abstract: i X We study the autocorrelation function of different types of eigenfunctions in quantum me- r chanical systems with either chaotic or mixed classical limits. We obtain an expansion of the a autocorrelation function in terms of the correlation length. For localized states, like bounc- ing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales. 1E-mail address: a.backer@bristol.ac.uk 3E-mail address: roman.schubert@physik.uni-ulm.de 1 1 Introduction The behavior of a quantum mechanical system in the semiclassical limit strongly depends on the ergodic properties of the corresponding classical system. In particular, the eigenfunctions semi- classically reflect the phase space structure of the classical system and therefore they depend strongly on wheth