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
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(6.40) |
Here
denotes the von Neumann density function, i.e.,
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(6.41) |
This transformation can be conveniently split into two steps.
For this we define:
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(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
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(6.43) |
Calculation of the Wigner distribution involves as a first step a
change to new
coordinates and
.
This gives
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(6.44) |
In a second step this is followed
by a Fourier transformation with respect to .
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(6.45) |
An inverse Fourier transform gives the com distribution from the Wigner distribution:
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(6.46) |
Instead of the momentum coordinate it is possible to introduce
the wave vector
which eliminates
from the
transformation:
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(6.47) |
Note that as is self-adjoint,
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]
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(6.48) |
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