Fluctuations of wavefunctions about their classical average

Abstract

Quantum-classical correspondence for the average shape of eigenfunctions and the local spectral density of states are well-known facts. In this paper, the fluctuations that quantum mechanical wave functions present around the classical value are discussed. A simple random matrix model leads to a Gaussian distribution of the amplitudes. We compare this prediction with numerical calculations in chaotic models of coupled quartic oscillators. The expectation is broadly confirmed, but deviations due to scars are observed.

Quantum chaos and thermalization in isolated systems of interacting particles

“…The results presented in Refs. [64,65,66,67,68,70,69,72,71,73] give evidence for a quantum-classical correspondence of the strength function. In the classical limit, the meaning of the strength function is just a projection of the energy surface of H 0 onto that of H, the fact that simplifies the analysis of quantum systems having a well defined classical limit.…”

“…The quantum-classical correspondence for the strength functions, as well as for the envelopes of eigenstates, has been thoroughly studied in Refs. [64,65,66,67,68,69,70,71,72,73,73] for various models of interacting particles.…”

“…As for the second condition, to justify the absence of strong correlations in the structure of eigenstates is not a simple task. Instead, a semi-analytical approach has been suggested [156,65,64,68,67,70,69,73,42,43] that allows one to predict the conditions under which chaotic eigenstates emerge depending on the model parameters. This approach is based on the concept of energy shell with respect to which the structure of eigenstates should be compared.…”

Semiclassical Husimi Function of Simple and Chaotic Systems

“…They showed that the Hamiltonians eigenfunctions of chaotic systems exhibit "scars" around unstable periodic orbit. An question that appears from those analyses is related to chaotic manifestation of classical chaos over the eigenfunctions in terms of quantities that are base independent [12][13][14][15]. In opposition, it has been reported that scars can exist in regions where there are no periodic orbits [16].…”

Parametric evolution of eigenstates: Beyond perturbation theory and semiclassics

“…For many years the study of parametric evolution for canonically quantized systems was restricted to the exploration of the crossover from integrability to chaos [7,8]. Only later [6] it has been realized that a theory is lacking for systems that are chaotic to begin with.…”