Quantum Phase Space Distribution Functions

Many ways for obtaining a two-variable function

from a one-variable one have been proposed ...

G. TORRES-VEGA

J. Chem. Phys. 98 (1993) 7040.

How can a quantum mechanical dynamical system be compared with its counterpart in classical mechanics? This is one of the principal questions in the theory of quantum chaos. Typically one studies quantum systems corresponding to classical dynamical systems which are known to be -- using the language of nonlinear dynamics -- weakly or hard chaotic. The object of these studies is to compare the outcomes of the two dynamical theories (classical and quantum mechanics) as effectively as possible, for example in order to understand which quantum properties correspond to certain classical phenomena.

In classical mechanics the concept of phase space has been used very successfully for a long time, especially when analyzing the dynamics of a system. Using this concept, the state of the system is represented by a point -- or a distribution of points -- in a space that is spanned by both the position and the momentum coordinates. In quantum mechanics, on the other hand, the state of the system can for example be formulated in terms of either the position or the momentum representation, where the wave functions (which are fields, as opposed to the points in classical phase space) exclusively depend on the position or momentum coordinates, but not on both sets of variables at the same time. Among other consequences this means that in the framework of a typical quantum mechanical representation one needs only half the number of dynamical variables as compared with the corresponding classical phase space. This raises the question of how these two apparently so different concepts of representing a state can be compared with each other.

The distribution function introduced by WIGNER in 1932 is an important tool for the investigation of this correspondence problem [Wig32]. Using the WIGNER distribution function, it is possible to describe quantum mechanical phenomena in a language which is as classical as possible, by employing a suitable quantum analogue of classical phase space. In the classical limit $\hbar=0$ the quantum mechanical phase space turns into its classical counterpart, and the distribution function defined in quantum phase space becomes the classical LIOUVILLE probability density in the same limit. This correspondence of the classical and quantum concepts paves the way for a straightforward comparison of the results of the two dynamical theories. In BERRY's words: ``WIGNER's picture is peculiarly well suited to the present problem, because it is in phase-space that the distinction between classical regular and irregular motion manifests itself most clearly, and so one can hope that the analogous quantal distinctions will reveal themselves with corresponding clarity in the form of [quantum phase space distribution functions] ...'' [Ber83].

The above reference to the LIOUVILLE probability density also indicates what interpretation one is aiming at when introducing quantum distribution functions: They are supposed to serve as quantum replacements of the classical phase space probability density. A number of different terms are used -- more or less synonymously -- for quantum distribution functions in order to emphasize this interpretation: (quasi-) probability distribution, (quasi-) probability density, distribution function, phase space distribution. The significance of the prefix quasi is discussed in section A.5.

Following the WIGNER distribution function many more (and mostly quite different) quantum distribution functions have been introduced since 1932; the most important ones of these are due to KIRKWOOD [Kir33], GLAUBER [Gla63b,Gla65] and HUSIMI [Hus40]. COHEN has developed a systematic classification of most of these distribution functions [Coh66];^A.1comprehensive accounts of this theory can be found in [BJ84,HOSW84,Lee95]. For the following discussion I mainly use the terminology of [Lee95].

For simplicity of notation I consider one-dimensional systems and pure states only, i.e. states which can be described by the specific density operator ${\hat{\rho}}(t)=\left\vert \psi(t) \right>\left< \psi(t) \right\vert$. Most of the following statements can also be formulated for mixed states described by arbitrary density operators, and many of the results can be generalized to higher dimensions as well.

Footnotes

...Cohen1966b;^A.1

An entirely different approach to quantum distribution functions is described in [TV93a,TV93b,TVF93,TVZ⁺96]. In the present study I do not consider this particular variant of the theory.