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arXiv:cond-mat/0608606 · Journal of Physics A Mathematical and Theoretical

Fisher information, delocalization and the semiclassical description of molecular rotation

Sergio Curilef, F Pennini, A Plastino, G L Ferri

Published Apr 24, 2007• 6 citations
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Abstract

We discuss phase-space delocalization for the rigid rotor within a semiclassical context by recourse to the Husimi distributions of both the linear and the 3D-anisotropic instances. The Husimi function can be viewed in a variety of ways. In particular, we obtain it as the expectation value of the density operator in a suitable basis of coherent states. Our treatment is based upon the concomitant Fisher information measures. The pertinent Wehrl entropy is also investigated in the linear case.

DOI
10.1088/1751-8113/40/19/012

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We discuss phase-space delocalization for the rigid rotor within a semiclassical context by recourse to the Husimi distributions of both the linear and the 3D-anisotropic instances. The Husimi function can be viewed in a variety of ways. In particular, we obtain it as the expectation value of the density operator in a suitable basis of coherent states. Our treatment is based upon the concomitant Fisher information measures. The pertinent Wehrl entropy is also investigated in the linear case.
We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.
We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.
Semiclassical delocalization in phase space constitutes a manifestation of the Uncertainty Principle, one indispensable part of the present understanding of Nature and the Wehrl entropy is widely regarded as the foremost localization-indicator. We readdress the matter here within the framework of the celebrated semiclassical Husimi distributions and their associatedWehrl entropies, suitably қ-deformed. We are able to show that it is possible to significantly improve on the extant phase-space classical-localization power.
We are concerned with an information theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase space probability (z,m I rho Iz,m) which is known as Husimi function, where rho is a density operator and Iz,m) are coherent states attached to an Euclidean mth Landau level. We obtain the Husimi function Qmbeta of the thermal density operator rhobeta of the harmonic oscillator, which leads, by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of Qmbeta such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions Qmbeta, m=0,1,... For m=1,2,..., we establish an exact expression for the Wehrl entropy of the density operator rhobeta and we discuss the behavior of this entropy with respect to the temperature parameter T = 1/beta.