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Monte Carlo Methods and Applications

A random cloud algorithm for the Schrödinger equation

Markus Kraft, Wolfgang Wagner

Published Nov 4, 2017• 0 citations

Abstract

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.

DOI: 10.1515/mcma-2017-0118

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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 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.

The quantum Liouville equation in the Wigner representation is solved numerically by using Monte Carlo methods. For incremental time steps, the propagation is implemented as a classical evolution in phase space modified by a quantum correction. The correction, which is a momentum jump function, is simulated in the quasi-classical approximation via a stochastic process. In this paper the technique, which is developed and validated in two- and three-dimensional momentum space, extends an earlier one-dimensional work. Also, by developing a new algorithm, the application to bound state motion in an anharmonic quartic potential shows better agreement with exact solutions in two-dimensional phase space. I. Introduction Nuclear interaction theory is formulated in the language of quantum mechanics, and hence the development of general methods of solutions to quantum dynamics will provide useful tools for application to a large class of problems in nuclear many-body theory. Different approaches...

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 method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.

A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed.