Comparing Wigner, Husimi and Bohmian distributions: Which one is a true probability distribution in phase space?

TL;DR

This paper compares Wigner, Husimi, and Bohmian phase space distributions, highlighting that Bohmian distribution uniquely provides a true probability distribution consistent with quantum charge and current densities.

Contribution

It demonstrates that Bohmian distribution is a positive, true probability distribution that accurately reproduces quantum charge and current densities, unlike Wigner and Husimi functions.

Findings

Bohmian distribution is positive and reproduces charge and current densities.

Wigner distribution can be negative in some regions.

Husimi distribution is positive but does not reproduce charge and current densities.

Abstract

The Wigner distribution function is a quasi-probability distribution. When properly integrated, it provides the correct charge and current densities, but it gives negative probabilities in some points and regions of the phase space. Alternatively, the Husimi distribution function is positive-defined everywhere, but it does not provide the correct charge and current densities. The origin of all these difficulties is the attempt to construct a phase space within a quantum theory that does not allow well-defined (i.e. simultaneous) values of the position and momentum of an electron. In contrast, within the (de Broglie-Bohm) Bohmian theory of quantum mechanics, an electron has well-defined position and momentum. Therefore, such theory provides a natural definition of the phase space probability distribution and by construction, it is positive-defined and it exactly reproduces the charge and current densities. The Bohmian distribution function has many potentialities for quantum problems, in general, and for quantum transport, in particular, that remains unexplored.