The expectation value of an operator in a state is given by performing the trace operation:

(6.52) |

In phase space the trace becomes a simple integral over the ordinary product of the corresponding Weyl transforms:

(6.53) |

Therefore, the computation of average values takes the same form as in classical statistical mechanics, with the Weyl transform and the Wigner function playing the roles of the classical observable and the classical probability distribution respectively.

or projection leads to marginal probability densities: a space like shadow or else a momentum-space shadow . Both are (bona-fide) probability densities, being positive semidefinite, as they are the expectation values of the projection operators on the corresponding eigenstates.

But neither can be conditioned on the other, as the uncertainty principle is fighting back: The Wigner function itself can, and most often becomes negative in some areas of phase space. In fact, the only pure-state Wigner function which is non-negative is the Gaussian.

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