The Wigner distribution function is a quasi-probability distribution function used to study
quantum mechanics in
phase space.
1Definition
The
Wigner Transform maps a quantum operator
to a classical function in phase space. Eugene Wigner used this to associate to a quantum system a real phase space function, now called Wigner distribution function (WDF)
[1]. It is defined to be the symbol associated to the
density operator describing the quantum system
In the case of a mixed state, the density operator can be written as convex combination of the pure state
wavefunctions For mixed states the WDF thus reads
while for a pure state it reduces to
2Properties
The WDF has many similarities to the classical distribution function:
is a real function, as follows by taking the complex conjugate and changing variables
. It is normalized to
in the following sense
When integrated over all momenta, it gives the probability density:
whereas when integrated over all positions, it yields the momentum distribution
where the subscripts
denote the dependence on
. The WDF thus has the attractive property that the marginal distributions, obtained by integrating over either the position or momentum variables, do reproduce the correct non-negative position and momentum probability distributions respectively, as specified by quantum mechanics.
Compared to the classical distribution function, the Wigner distribution function has the peculiar property that it may assume negative values. For this reason it is usually called a
quasi-probability distribution and cannot be interpreted as a phase space probability density in the sense of
classical mechanics.
The dynamical evolution of the Wigner distribution function is described by the
Wigner equation, reducing to the classical
Vlasov equation in the formal
limit.
References:- [1]^ E.P. Wigner: On the quantum correction for thermodynamic equilibrium. In: Phys. Rev. 40, 1932, pp. 749–759.