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Zeitschrift für Naturforschung A

Applications d'une representation de la mecanique quantique par des fonctions sur l'espace de phase

P. Huguenin

Published Jul 1, 1973• 8 citations
Publisher

Abstract

The Weyl-Wigner transformation enables us to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined by the quasiprobability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix.

DOI
10.1515/zna-1973-0708

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The Weyl-Wigner transformation enables us to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined by the quasiprobability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix.
The Weyl-Wigner transformation enables one to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined bv the quasi-probability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix. (GE)
We propose a Wigner transformation of the density matrix and operators of a system with bounded coordinates and discrete momentum spectra (typically, a particle in a box) and prove that all requirements for such a transformation are satisfied. The phase-space mapping adopted here is employed to compute the quasiprobability distributions associated with the pure states of a quantum rotator and the image of the angular momentum operator. The mapping of the dynamics of these objects, both in the reversible and irreversible frame, is constructed as well, and the conditions for the existence of a classical limit in the form of a Fokker-Planck equation of motion are investigated.
Les expressions locales d'une demi-forme Lagrangienne sur une sousvariete Lagrangienne quantifiee de l'espace des phases sont les fonctions d'onde de la mecanique quantique. Nous montrons que le transport de ces demi-formes par le flot associe a un Hamiltonien H permet de retrouver les solutions asymptotiques de l'equation de Schrodinger donnes par la formule de Maslov. Nous etudions ensuite le cas ou l'on remplace ce flot par celui associe au Bohmien, et arrivons a la conclusion que l'emploi des demi-formes Lagrangiennes permet d'elaborer une mecanique quantique en espace de phase.
Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (e.g. quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides — coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.
We continue the analysis of relativistic phase space, identified with the quotient of the Poincare group in 1+1 dimensions by the time translation subgroup. Proceeding by contraction from the corresponding (Anti) de Sitter group SO 0 (2, 1), we obtain a realisation of the Poincare representation P(m) of mass m in a space of functions defined on phase space. The contraction singles out a privileged section in the Poincare group, with a unique left and right invariant measure. Using that section, we show that the representation P(m) is square integrable over the coset space, i.e. phase space. From this we build a new set of Poincare coherent states, and more generally weighted coherent states, which have all the usual properties, resolution of the identity, overcompleteness, reproducing kernel, orthogonality relations. Finally we derive the corresponding Wigner transform On poursuit l'analyse de l'espace de phase relativiste, identifie au quotient du groupe de Poincare en dimension 1+1 par le sous-groupe des translations temporelles. Par contraction a partir du groupe Anti-de Sitter correspondant SO 0 (2, 1), on obtient une realisation de la representation P(m) de masse m du groupe de Poincare dans un espace de fonctions definies sur l'espace de phase. La contraction selectionne une section privilegiee dans le groupe de Poincare, possedant une unique mesure invariante a la fois a gauche et a droite. Avec ce choix de section, on montre que la representation P(m) est de carre sommable sur l'espace quotient, c'est-a-dire l'espace de phase. A partir de la, on construit une nouvelle famille d'etats coherents de Poincare et, plus generalement, d'etats coherents avec poids, qui ont toutes les proprietes usuelles: resolution de l'identite, surcompletude, noyau reproduisant, relations d'orthogonalite. Enfin, on derive la transformation de Wigner correspondante
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.