Wigner functions, signed particles, and the harmonic oscillator

Abstract

In this paper, we introduce the simple harmonic oscillator and we address it in the Wigner formulation of quantum mechanics, therefore describing the whole problem in terms of quasi-distribution functions defined over the phase-space. The harmonic oscillator represents a very important problem as it provides exact solutions in both stationary and transient regimes. Subsequently, we outline the time-dependent signed particle Wigner Monte Carlo method and simulate the oscillator problem starting from stationary initial conditions, i.e. rotationally invariant functions in the phase-space, showing no evolution in time of the distribution function as expected. This work is, thus, twofold. On the one hand, one may see it as a short review effort to demonstrate the convenience of utilizing a phase-space approach in this particular context, suggesting that it could be the case again for different interesting problems. On the other hand, it represents a further opportunity to validate the signed particle Monte Carlo method, showing that a new reliable and powerful tool is available for the time-dependent simulation of quantum systems.

A study of entangled systems in the many‐body signed particle formulation of quantum mechanics

“…The reader should consider that in this particular case, discernible from the postulates above, the signed particle formulation corresponds to the many‐body Wigner Monte Carlo method described in Ref. [ ]. Indeed, the adoption of this version of Postulate II in the current work (and as a consequence, the many‐body Wigner Monte Carlo method), provides certain advantages, which require further investigation.…”

“…Notably, the restriction of the formalism to finite domains and semidiscrete phase‐spaces provides a correspondence to the well‐known signed particle Wigner Monte Carlo method. The latter has been thoroughly validated in the past few years, augmenting it to the case of time‐dependent, ab initio (or first‐principle) simulations of quantum many‐body problems …”

On the hydrogen atom beyond the Born-Oppenheimer approximation

“…This section introduces the signed particle formulation of quantum mechanics recently suggested in Ref. [4]. It consists of a set of three postulates which are reported below.…”

“…This approach is completely equivalent to other (more standard) formulations of quantum mechanics such as, for example, the one suggested by E. Schrödinger (in other words, they provide exactly the same predictions). Importantly, the formulation has been extended to time‐dependent, quantum many‐body problems showing already to be extremely promising and useful, for example, in the case of indistinguishable Fermions …”

“…[2,3] Recently, a new formulation of quantum mechanics based on the concept of signed particles has been introduced. [4] The underlying principle of the formulation utilizes ensembles of classical signed field-less particles in the simulation of time-dependent quantum system. The formalism consist of a set of three postulates, which describe the evolution of a quantum system.…”