An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism

The Wigner formulation of quantum mechanics is a very intuitive approach which allows the comprehension and prediction of quantum mechanical phenomena in terms of quasidistribution functions. In this review, our aim is to provide a detailed introduction to this theory along with a Monte Carlo method for the simulation of time-dependent quantum systems evolving in a phase-space. This work consists of three main parts. First, we introduce the Wigner formalism, then we discuss in detail the Wigner Monte Carlo method and, finally, we present practical applications. In particular, the Wigner model is first derived from the Schrödinger equation. Then a generalization of the formalism due to Moyal is provided, which allows to recover important mathematical properties of the model. Next, the Wigner equation is further generalized to the case of manybody quantum systems. Finally, a physical interpretation of the negative part of a quasidistribution function is suggested. In the second part, the Wigner Monte Carlo method, based on the concept of signed (virtual) particles, is introduced in detail for the singlebody problem. Two extensions of the Wigner Monte Carlo method to quantum manybody problems are introduced, in the frameworks of time-dependent density functional theory and ab-initio methods. Finally, in the third and last part of this paper, applications to singleand many-body problems are performed in the context of quantum physics and quantum chemistry, specifically focusing on the hydrogen, lithiumand boron atoms, theH2 molecule and a system of two identical Fermions. We conclude this work with a discussion on the still unexplored directions the Wigner Monte Carlo method could take in the next future. © 2015 Elsevier B.V. All rights reserved.