Probabilistic Particle Models of Classical and Quantum Wave Systems

Abstract

In this article, we present particle models of classical dissipative wave and quantum dispersive wave systems. A particle model of classical waves is designed such that the probability density function in terms of the position of a particle moving in discrete space evolves equivalently to the wave propagation of the original system. The particle model of dissipationless wave systems is applied to quantum digital wave filters in which binary signals are represented by single-electrons or single-flux quanta. Wave systems with dissipation are considered as diffusion systems with damped oscillation. Since a diffusion system is a set of random walkers, the model is utilized also as a parallel pseudorandom sequence generator. The autocorrelation of the sequences is set to be positive or negative depending on the model parameter. The quantum waves to be modeled in this article are nonrelativistic and described by the Schrödinger or Pauli equation. The probability density function of the particle models is governed by the Fokker-Planck equation, which is stochastically equivalent to the Schrödinger or Pauli equation. Then, the Langevin equation possessing the same drift term and diffusion coefficient as the Fokker-Planck equation is obtained. Since the Langevin equation, a stochastic ordinary differential equation, describes the behavior of the modeled particles, the modeling provides the basis for constructing circuit simulator models of quantum devices.