Theory and application of the quantum phase-space distribution functions

Thomas Curtright, COSMAS K. ZACHOS

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple - indeed, classical - for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.

Fredrik Zachariasen

J. E. Moyal

An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of dynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to specifying a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum kinematics . In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of quantum dynamics and found to be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative to the Schrödinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calculation of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics , deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verification are discussed.

Yee Kin Tsui

Kazuo Takatsuka, Hiroki Nakamura

A new semiclassical theory is formulated by introducing a new distribution function to be propagated in phase space. Primarily this theory is intended to describe molecular collisions such as reactive scattering, but is also applicable to intramolecular processes including bound state problems. This new distribution function is a kind of characteristic function in the terminology of the phase-space-quantum theory developed by Weyl, Wigner, Moyal and others, and is called a dynamical characteristic function (DCF). The equation of motion and some properties of the DCF are presented. The essential feature of our theory is that the semiclassical DCF can be propagated using classical trajectories. This distribution function carried information not only on the density, but also on the phase (action integral). In this sense this theory can be regarded as a kind of phase-space-path-integral formalism. The semiclassical approximation is analyzed from the viewpoint of path integrals. The Wigner distribution function is also discussed in the context of our analysis. As a theoretical application, the semiclassical DCF is shown to reproduce the expression of the periodic orbit theory for bound states. Also, as a simple numerical application, a propagation of the DCF on a nonquadratic (Morse) potential is presented.