6 - The Wigner Distribution Function: Analytical Evaluation

Abstract

This chapter discusses and evaluates the Wigner distribution function. The Wigner distribution function was introduced as the simplest quantum analog of the classical phase space distribution function permitting to find probabilities and expectation values of quantum operators, using as much classical language and methods as allowed. The Wigner method establishes a rule to associate a number function in phase-space with every operator being a function of position and momentum operators. The expectation values of quantum mechanical observables can be calculated in the same mathematical form as the averages of tile classical statistical mechanics rather than through the operator formalism of quantum mechanics. The dynamical state of the particle is described by specifying at each time the wave function, which determines the probability density of the position coordinate. A primary illustration of the valence of the Wigner distribution function to convey an intermediate signal description between the pure space and pure spatial-frequency descriptions is provided. The marginal distributions and inversion formulas are also elaborated.