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
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.1}comprehensive 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
. 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.

- ...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.

- Definition of Quantum Phase Space Distribution Functions
- Special Distribution Functions
- Minimum Uncertainty States and the HUSIMI Distribution Function

- Dynamics
- On the Interpretation as Probability Densities
- Typical Applications