6.2.4 Probabilistic Structure

The standard semantics of quantum mechanics as specified by the Kopenhagen interpretation is a probabilistic one. The probabilistic structure manifests itself as inner product in a Hilbert space: The space of operators with the trace as inner product is a Hilbert space. The operator product has already been translated to phase space in the preceding section. Now we come to the translation of the inner product.

The expectation value $\bar{A}$ of an operator $A$ in a state $\rho$ is given by performing the trace operation:

$$\bar{A} = {\mathrm{tr} (}A \rho) .$$ (6.52)

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

$${\mathrm{tr} (}({A} {\bf {\rho}}) = \frac{1}{2\pi\hbar}\int \int A_w(x,p) f(x,p) dp dx .$$ (6.53)

Therefore, the computation of average values takes the same form as in classical statistical mechanics, with the Weyl transform $A_w$ and the Wigner function $f$ playing the roles of the classical observable $A(x,p)$ and the classical probability distribution $f^{cl}$ respectively.

$p-$ or $x-$projection leads to marginal probability densities: a space like shadow $X(x) = \int f(x,p) dp$ or else a momentum-space shadow $P(p) = \int f(x,p) dx$. 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.