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## 6.2.2 Definition of Wigner Function

Among the several transforms of the wave function or in general of the density operator, that provide a phase space view of the quantum state, the Wigner function is the one that has found more applications, mainly in statistical mechanics, but also in quantum chemistry, molecular dynamics, scattering theory, or quantum optics.

The Wigner function is (up to a constant factor) the Weyl transform of the quantum-mechanical density operator. For a particle in one dimension it takes the form (6.40)

Here denotes the von Neumann density function, i.e., (6.41)

with the density operator.

This transformation can be conveniently split into two steps. For this we define:  (6.42)

We call the center of mass coordinate and the distance coordinate. These names should not be physically misleading as the von Neumann equation does not describe a two-particle problem but quantum correlations.

The inverse relation is  (6.43)

Calculation of the Wigner distribution involves as a first step a change to new coordinates and . This gives (6.44)

We call the center of mass (com) distribution.

In a second step this is followed by a Fourier transformation with respect to . (6.45)

An inverse Fourier transform gives the com distribution from the Wigner distribution: (6.46)

Instead of the momentum coordinate it is possible to introduce the wave vector which eliminates from the transformation: (6.47)

Note that as is self-adjoint, we have and So the Wigner distribution is real. However, in contrast to its classical analogon it can assume negative values. As position and momentum operators do not commute and hence cannot be measured together, this does not present an inconsistency.

The Wigner distribution possesses an important - duality given by the alternative definition [FJP03] (6.48)

Here denotes the density function in momentum representation, that is: where denotes the eigenvector of the momentum operator with eigenvalue .