Boundaries and profiles in the Wigner formalism

TL;DR

This paper addresses issues in the Wigner formalism for quantum devices by proposing a correct dynamical equation that incorporates non-local potential effects and boundary conditions to avoid non-physical solutions.

Contribution

It introduces a revised dynamical equation for the Wigner function that accounts for external potential influences and boundary effects, improving the physical accuracy of the model.

Findings

Correct dynamical equation reduces non-physical solutions

Inclusion of non-local potential effects is essential

Careful boundary condition selection prevents non-physical results

Abstract

We consider a quantum device contained in an interval in the context of the Weyl-Wigner formalism. This approach was originally suggested by Frensley, and is known to be plagued with several problems, such as non-physical and non-unique solutions. We show that some of these problems may be avoided if one writes the correct dynamical equation. This requires the (non-local) influence of the potential outside of the device and the inclusion of singular boundary potentials. We also discuss the problem of imposing boundary conditions on the Wigner function that mimic the effect of the external environment. We argue that these conditions have to be chosen with extreme care, as they may otherwise lead to non-physical solutions.