Spectral and entropic characterizations of Wigner functions: Applications to model vibrational systems

Abstract

The Wigner function for the pure quantum states is used as an integral kernel of the non-Hermitian operator K, to which the standard singular value decomposition (SVD) is applied. It provides a set of the squared singular values treated as probabilities of the individual phase-space processes, the latter being described by eigenfunctions of KK(+) (for coordinate variables) and K(+)K (for momentum variables). Such a SVD representation is employed to obviate the well-known difficulties in the definition of the phase-space entropy measures in terms of the Wigner function that usually allows negative values. In particular, the new measures of nonclassicality are constructed in the form that automatically satisfies additivity for systems composed of noninteracting parts. Furthermore, the emphasis is given on the geometrical interpretation of the full entropy measure as the effective phase-space volume in the Wigner picture of quantum mechanics. The approach is exemplified by considering some generic vibrational systems. Specifically, for eigenstates of the harmonic oscillator and a superposition of coherent states, the singular value spectrum is evaluated analytically. Numerical computations are given for the nonlinear problems (the Morse and double well oscillators, and the Henon-Heiles system). We also discuss the difficulties in implementation of a similar technique for electronic problems.

Harmonium as a laboratory for mathematical chemistry

“…Currently Gill and coworkers [14] pursue a vigorous program to understand correlation energies via Wigner intracules: our method also sits transversally midway between the 6-variable Gilbert and Gill programs. We mention as well a different fusion of the Schmidt and Wigner ways, related to information theory [15], and a recent quantum phase space view of harmonium by Dahl [16].…”