Everything about Wigner-ville Distribution totally explained
» See also Wigner distribution, a disambiguation page.The
Wigner quasi-probability distribution (also called the
Wigner function or the
Wigner-Ville distribution) is a special type of
quasi-probability distribution. It was introduced by
Eugene Wigner in 1932 to study
quantum
corrections to classical statistical mechanics. The goal was to supplant the
wavefunction that appears in
Schrödinger's equation with a probability distribution in
phase space.
It is a
generating function for all spatial
autocorrelation functions of a given quantum-mechanical wavefunction ψ(x).
Thus, it maps on the quantum
density matrix in the map between real phase-space functions and Hermitian operators introduced by
Hermann Weyl in
1927, in a context related to
representation theory in mathematics (cf.
Weyl quantization in physics). In effect, it's the
Weyl-Wigner transform of the density matrix, so the realization of that
operator in phase space. It was later rederived by J. Ville in
1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal.
In
1949,
José Enrique Moyal, who had also rederived it independently, recognized it as the quantum moment-generating functional, and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (cf
Weyl quantization). It has applications in
statistical mechanics,
quantum chemistry,
quantum optics, classical
optics and signal analysis in diverse fields such as
electrical engineering,
seismology,
biology, speech processing, and engine design.
Relation to classical mechanics
A classical particle has a definite position and momentum, and hence it's represented by a point in phase space. Given a collection (
ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation fails
for a quantum particle, due to the
uncertainty principle. Instead, the above quasi-probability Wigner distribution plays an analogous role, but doesn't satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions.
For instance, the Wigner distribution can and normally does go negative for states which have no classical model---and is a convenient indicator of quantum mechanical interference.
Smoothing the Wigner distribution through a filter of size larger than
(for example, convolving with a
phase-space Gaussian to yield the Hussimi representation, below), results in a positive-semidefinite
function, for example, it may be thought to have been coarsened to a
semi-classical one.
Regions of such negative value are provable (by convolving them
with a small Gaussian) to be "small":
they can't extend to compact regions larger than a few
,
and hence disappear in the classical limit. They are shielded
by the
uncertainty principle, which doesn't allow precise
location within phase-space regions smaller than
,
and thus renders such "negative probabilities" less paradoxical.
Mathematical definition
The Wigner distribution
P(
x,
p) is defined as:
»
The inverse of this transformation is called the
Weyl transformation, not to be confused with another definition of the
Weyl transformation. The
Wigner function is the Weyl-Wigner Transform of the
density matrix.
Other related quasi-probability distributions
See Quasi-probability distribution for more detail.
The Wigner distribution was the first quasi-probability distribution, but many more followed, formally equivalent and
transformable to and from it. As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications:
Glauber P representation
Husimi Q representation
Historical note
As indicated, the formula for the Wigner function was independently derived several times in different contexts. In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac, albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom. Incidentally, Dirac would later become Wigner's brother-in-law. Symmetrically, in most of his legendary 18-month correspondence with Moyal in the mid 1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.
See references.
Further Information
Get more info on 'Wigner-ville Distribution'.
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