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arXiv:cond-mat/0312680 · Physical Review E

Heisenberg-Fisher thermal uncertainty measure

F. Pennini, Angel Ricardo Plastino

Published May 21, 2004• 40 citations

Abstract

We establish a connection among (i) the so-called Wehrl entropy, (ii) Fisher's information measure I(beta), and (iii) the canonical ensemble entropy for the one-dimensional quantum harmonic oscillator (HO). We show that the contribution of the excited HO spectrum to the mean thermal energy is given by I(beta), while the pertinent canonical partition function is essentially given by another Fisher measure: the so-called shift invariant one. Our findings should be of interest in view of the fact that it has been shown that the Legendre transform structure of thermodynamics can be replicated without any change if one replaces the Boltzmann-Gibbs-Shannon entropy by Fisher's information measure [Phys. Rev. E 60, 48 (1999)]]. Fisher-related uncertainty relations are also advanced, together with a Fisher version of thermodynamics' third law.

DOI: 10.1103/physreve.69.057101

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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.

The Wehrl entropy or the "classical" entropy associated with a quantum system is the entropy of the probability distribution in phase space, corresponding to the Husimi Q-function in terms of coherent states.In the present paper, we shall focus our attention on the examination of the Wehrl entropy for both the pure and the mixed (thermal) states of the pseudoharmonic oscillator (PHO).The choice of the PHO is interesting because this oscillator is an intermediate between the ideal one-dimensional harmonic oscillator (HO-1D) and the more practical anharmonicone.