Since
,
for each ,
contains exactly the same information as
the HILBERT space vector
it is possible to give a
complete formulation of quantum mechanics by means of phase space
distribution functions only, that is without referring to
quantum
state vectors or
wave functions. In such a framework describes the state of
the system at time , whereas the time evolution of is
determined by the equation of motion

which takes the role normally occupied by the SCHRÖDINGER equation in the conventional formulation of quantum mechanics. Here, is the

As an example, for the special case of the WIGNER distribution function
equation (A.80) becomes

Obviously, the equations of motion (A.80) and (A.81) are quite awkward to work with. This is the most important reason why in practical applications quantum mechanics hardly ever is discussed with distribution functions completely replacing quantum states and wave functions. Rather, even when the final task is to obtain distribution functions, typically the well-established version of quantum mechanics is used, i.e. one starts by solving -- numerically, if necessary -- the SCHRÖDINGER equation and then inserts the solution into equation (A.13) or (A.74), for instance, in order to compute the desired .

But in addition to the computation of distribution functions, the equation
of motion for WIGNER's function is of importance for the
investigation of the
semiclassical approximation with
.
For Hamiltonians of the type
equation (A.81) can be ordered with respect to powers of
:

with as usual denoting terms that are at least quadratic in and that give rise to the quantum mechanical corrections to the classical phase space distribution function as obtained from the classical LIOUVILLE equation

Therefore in the classical limit the equation of motion (A.83) for formally becomes the LIOUVILLE equation (A.84); this makes WIGNER's function a useful tool for the investigation of the correspondence issue. Furthermore, for potentials that do not contain powers of exceeding 2, the dynamics of is completely classical, since in this case the equations (A.83) and (A.84) coincide. But note that itself depends on , such that the similarity of these quantum mechanical and classical equations of motion is formal in the first place, and the details of obtaining equation (A.84) from (A.83) in the classical limit are nontrivial in general.

I return to equations (A.82) and (A.83) in section A.5 when I discuss the interpretation of as a (quasi-) probability distribution function in quantum phase space.