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Journal of Modern Physics

Semiclassical Husimi Function of Simple and Chaotic Systems

Adélcio C. Oliveira

Published Jan 1, 2012• 7 citations
Publisher

Abstract

We review the semiclassical method proposed in [1], a generalization of this method for n-dimensional system is presented. Using the cited method, we present an analytical method of obtain the semiclassical Husimi Function. The validity of the method is tested using Harmonic Oscillator, Morse Potential and Dikie’s Model as example, we found a good accuracy in the classical limit.

DOI
10.4236/jmp.2012.38094

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We review the semiclassical method proposed in [1], a generalization of this method for n-dimensional system is presented. Using the cited method, we present an analytical method of obtain the semiclassical Husimi Function. The validity of the method is tested using Harmonic Oscillator, Morse Potential and Dikie’s Model as example, we found a good accuracy in the classical limit.
We extend the Husimi (coherent state) based version of linearized semiclassical theories for the calculation of correlation functions to the case of survival probabilities. This is a case that could be dealt with before only by use of the Wigner version of linearized semiclassical theory. Numerical comparisons of the Husimi and the Wigner case with full quantum results as well as with full semiclassical ones will be given for the revival dynamics in a Morse oscillator with and without coupling to an additional harmonic degree of freedom.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
The dynamics generated by non-Hermitian Hamiltonians are often less intuitive than those of conventional Hermitian systems. Even for models as simple as a complexified harmonic oscillator, the dynamics for generic initial states shows surprising features. Here we analyze the dynamics of the Husimi distribution in a semiclassical limit, illuminating the foundations of the full quantum evolution. The classical Husimi evolution is composed of two factors: (i) the initial Husimi distribution evaluated along phase-space trajectories and (ii) the final value of the norm corresponding to each phase-space point. Both factors conspire to lead to intriguing dynamical behaviors. We demonstrate how the full quantum dynamics unfolds on top of the classical Husimi dynamics for two instructive examples.