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.
Quantum mechanics Wigner formulation Monte Carlo methods Harmonic oscillator