Wigner quantum mechanics is reformulated in a discrete momentum space and analyzed with a Monte Carlo algorithm, in order to for solve integral equations and are thus associated with a particle picture. General quantum transport phenomena, accounting for transients, boundary conditions and initial conditions may be modeled in terms of quasi-particles involving attributes like drift, generation, sign, and annihilation in a phase space grid. The model is examined in an ultimate regime, where classical and quantum dynamics become equivalent. The difference between the classical and the quantum mean values of a physical quantity associated with the evolution of an initial state is negligible for up to quadratic potentials. In this case the commutator coincides with the Poisson bracket and the physical aspects are determined by the initial condition only. This ultimate parity may be used to setup benchmark experiments testing the properties of quantum computational approaches. In particular, in the phase space, the ballistic Boltzmann and coherent Wigner evolution become equivalent for linear potentials. Why has this duality not yet been used as a reference for validation of Wigner transport simulation methods? The reason is that the Wigner potential becomes a generalized function, namely a delta function derivative, which precludes any exact numerical treatment: even the standard, infinitely coherent in space definition of the Wigner potential diverges.
This research aims at both, the development of an asymptotic approach as well as validating of our Wigner particle model for this extreme case. The model, which entangles particle attributes such as generation, sign and annihilation at consecutive time steps, must resemble the effect of acceleration due to the applied electric field. The equivalence between the two pictures is achieved only at the limit of the coherence length approaching infinity, so that the finite case reveals both classical and quantum effects. The peculiarities of the transport in this asymptotic regime are analyzed within simulations, benchmarking the behavior of the Wigner function.