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Kinetic and Related Models

A random cloud model for the Schrödinger equation

Wolfgang Wagner

Published Jan 1, 2014• 14 citations
Publisher

Abstract

The paper is concerned with the construction of astochastic model for the spatially discretizedtime-dependent Schrödinger equation.The model is based on a particle system witha Markov jump evolution.The particles are characterized by a sign (plus or minus),a position (discrete grid) and a type (real or imaginary).The jumps are determined by the creation of offspring.The main result is the construction of a family of complex-valued randomvariables such that their expectedvalues coincide with the solution of theSchrödinger equation.

DOI
10.3934/krm.2014.7.361

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The paper is concerned with the construction of astochastic model for the spatially discretizedtime-dependent Schrödinger equation.The model is based on a particle system witha Markov jump evolution.The particles are characterized by a sign (plus or minus),a position (discrete grid) and a type (real or imaginary).The jumps are determined by the creation of offspring.The main result is the construction of a family of complex-valued randomvariables such that their expectedvalues coincide with the solution of theSchrödinger equation.
Abstract A class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.
A class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.
Abstract In this paper we present a numerical scheme for the Random Cloud Model (RCM) on a bounded domain which approximates the solution of the time-dependent Schrödinger equation. The RCM is formulated as a Markov jump process on a particle number state space. Based on this process a stochastic algorithm is developed. It is shown that the algorithm reproduces the dynamics of the time-dependent Schrödinger equation for exact initial conditions on a bounded domain. The algorithm is then tested for two different cases. First, it is shown that the RCM reproduces the analytic solution for a particle in a potential well with infinite potential. Second, the RCM is used to study three cases with finite potential walls. It is found that the potential triggers processes, which produces RCM particles at a high rate that annihilate each other.
The principle of "indirect continuous measurement" in "open quantum system theory" is usually described by non-usual types of stochastic differential equations. These equations are called "stochastic Schrödinger equations" and their solutions are called "quantum trajectories". Physically, they describe the random evolution of the state of a quantum system undergoing indirect quantum measurement (such models are widely used in quantum optics, quantum computing and quantum information theory). In this chapter, we consider a physically realistic discrete-time setup for two-level quantum systems and we present the theory of "discrete quantum trajectories". These discrete trajectories are Markov chains which can be expressed as solutions of "discrete-time" stochastic differential equations". In particular, these equations appear as time discretization of "stochastic Schrödinger equations". Going to the continuous-time limit, we justify the stochastic Schrödinger equations associated to the two-level systems. Within this approach, we obtain two different types of behaviors described either by jump-type or diffusive-type stochastic differential equations. Finally we investigate the large time behavior of the solutions and we prove return to equilibrium properties for the associated physical models.
The one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrödinger equation determines the model of motion of particles in the nnn-quantum state as orbits of a stationary random walk at the moment of jump from the n−1n-1n−1-qu