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Physica D Nonlinear Phenomena

Correlation function behavior in quantum systems which are classically chaotic

G. P. Berman, Andrey R. Kolovsky

Published Jul 1, 1983• 27 citations
Publisher
DOI
10.1016/0167-2789(83)90313-5

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The evolution of wave packets under the influence of a Hénon–Heiles potential has been investigated by direct numerical solution of the time-dependent Schrödinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈ψ(0)‖ψ(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.
The problems of chaos and relaxation have a fundamental importance in the study of many-body classical and quantum systems. We investigate some of the issues related to these problems numerically in classical and quantum spin systems. New results reported in this thesis include: (i) A remarkably simple algorithm for discriminating chaotic from nonchaotic behavior in classical systems using a time series of one macroscopic observable. The effectiveness of this algorithm stems from the qualitative differences in the power spectra of chaotic and nonchaotic systems. (ii) A modified version of the Onsager regression relation applicable to pure quantum states. (iii) An efficient algorithm for computing the infinitetemperature time correlation functions in systems with large Hilbert spaces. (iv) Absence of exponential sensitivity to small perturbations in macroscopic nonintegrable systems of spins 1/2 . Such a behavior is contrasted with the exponential sensitivity to small perturbations in chaotic classical spin systems. (v) Accurate numerical investigations of free induction decay and spin diffusion in certain spin lattices. The consequences of these results have implications for the foundations of statistical mechanics or practical problems such as computing the long-time behavior of the free induction decay in solids.