ABSTRACT
Given the wave function ψ(x, t) we compute the probability density function ρ(x, t) = |ψ(x, t)|2 and easily visualize the distribution. Knowing ψ(x, t) the distribution in momentum px is given by
σ(px, t) = |χ(px, t)|2 = ∣∣∣ 1√
2π
ψ(x, t)e−ikx dx ∣∣∣2 , (6.1)
where χ(px, t) is the momentum wave function. The momentum distribution is difficult to visualize from the given ψ(x, t). Thus, we wish to have a function that can display the probability distribution in the variables x and px simultaneously. Recall that, in classical mechanics phase space is used to visualize the solutions of systems. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture [1-8]. The idea is that every state of a quantum system is describable by a distribution (or function) on the classical phase space. In the phase space picture, starting from the Schrödinger wave function, we are able to construct the Wigner distribution function in terms of x and px. It does not represent a joint probability distribution for x and px because the uncertainty principle disallows the simultaneous determination of these variables with desired accuracy.
