Densities

The motivation of the theory outlined in this appendix was to construct distribution functions which can play the same role in quantum mechanics as the conventional phase space probability densities do in classical mechanics (cf. pages ff). Therefore it now needs to be checked to what extent the functions actually have the typical properties of probability densities.

A
``genuine''
phase space probability density is characterized by
the following properties:^{A.12}

- Reality: for all , and .
- Positivity: for all , and .
- By integration over and , must give the correct
quantum mechanical
*marginal probability densities*in position and momentum space, respectively:^{A.13}

The properties (i)-(iii)
are not automatically satisfied for all distribution functions
,
but must be checked for each individual .
They are by no means necessary consequences of the definitions in
sections A.1 and A.2.
On the contrary, it can be shown that for nonzero *none* of the
known satisfies all three conditions.
This may be interpreted as a consequence
of HEISENBERG's uncertainty relation and was already
observed by WIGNER in 1932 [Wig32].
This shortcoming of all
distribution functions
is the reason for using the prefix
*quasi* in terms like quasiprobability density or quasiprobability
distribution function.

WIGNER's distribution function, for example, is real-valued and gives the correct marginal probability distributions for position and momentum, but it can take on negative values. In fact, it typically oscillates very rapidly and with large amplitude between positive and negative values under variation of or . This behaviour of the WIGNER function is further discussed and demonstrated for some examples in section A.6.

For the antinormal-ordered and the HUSIMI distribution functions one has, using equations
(A.51, A.74),

respectively. Both are real-valued and non-negative. They are even
bounded and do not oscillate as rapidly as . But, on the
other hand, they do not yield the correct probability densities for
and by
marginalization.
Again I refer the reader to section
A.6 for more details.
Table A.1 gives an overview of the essential properties
of the distribution functions defined in sections A.2
and A.3.

It is important to keep in mind that despite the different properties of the distribution functions, they are all equivalent in the sense that all of them contain exactly the same information about the state of the system. This can be exploited by choosing that type of distribution function (respectively by choosing that kernel function ) that is most suited to study the particular problem in question.

For the purpose of comparing the classical and quantum mechanical
*equations of motion* of a given system, one usually uses WIGNER's
function (cf. equation (A.82); see [Bun95] for details).
is also especially well suited for the discussion of
scattering systems, because in such systems,
for large energies,
the classical limiting case
often is a good approximation to the exact quantum system already;
equation (A.82) can then be used to
determine
the
quantum
correction terms in a
systematic
way.
On the other hand, if the quantum and classical *phase space
(quasi-)
probability densities* themselves are in the focus of attention, then most
frequently the HUSIMI distribution function
or the coherent state representation
are employed.
In section A.6 I discuss some of the advantages of
this choice by considering two familiar quantum states as examples.

- ... properties:
^{A.12} - There are further, more technical conditions that are to be satisfied by a probability density [Lee95]; these additional properties are not needed for the present discussion.
- ... respectively:
^{A.13} -
Equations (A.85a, A.85b)
are just two specific
examples for the marginalization of a distribution function.
Generally speaking, a
*marginalization*of a distribution function is a line integral over in phase space. In equations (A.85a, A.85b), the line integrals are taken along the and axes of phase space, respectively. More on marginalizations of distribution functions may be found in [MMT97].