Formally, the phase space representation is closer to its classical counterpart than Schrödinger's formulation. The formulation of quantum mechanics in phase space is outside the scope of most introductory books on quantum mechanics. Here we present an introduction, which focuses on the Wigner distribution.
First we translate the abstract quantum mechanical machinery
(operators and probabilities) to the Wigner language.
Special
care is devoted to the notion of marginal distributions
and marginal expectation values.
We point out that these present an instance
of the operator ordering problem and their definition
is non-unique. We then
present a phase space distribution which has the
property that it is positive everywhere and gets
carrier, current and energy density
consistent with the standard definitions for
a Schrödinger wave function .
Finally we discuss the quantum mechanical analogon of the
classical notion of
trajectories and their peculiarities.