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.
Funding statement: This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its CREATE programme. Markus Kraft gratefully acknowledges the support of Weierstrass Institute for Applied Analysis and Stochastics and the support of the Alexander von Humboldt Foundation through the Friedrich Wilhelm Bessel Research Award.
References
[1] R. Becerril, F. S. Guzmán, A. Rendón-Romero and S. Valdez-Alvarado, Solving the time-dependent Schrödinger equation using finite difference methods, Rev. Mex. Fis. E 54 (2008), 120–132. Search in Google Scholar
[2] P. Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Texts Appl. Math. 31, Springer, New York, 2013. Search in Google Scholar
[3] D. O. Hayward, Quantum Mechanics for Chemists, Tutor. Chem. Texts 14, The Royal Society of Chemistry, London, 2002. Search in Google Scholar
[4] M. Kraft and W. Wagner, An improved stochastic algorithm for temperature-dependent homogeneous gas phase reactions, J. Comput. Phys. 185 (2003), 139–157. 10.1016/S0021-9991(02)00051-7Search in Google Scholar
[5] M. Kraft and W. Wagner, Numerical study of a stochastic particle method for homogeneous gas-phase reactions, Comput. Math. Appl. 45 (2003), no. 1–3, 329–349. 10.1016/S0898-1221(03)80022-6Search in Google Scholar
[6] O. Muscato and W. Wagner, A class of stochastic algorithms for the Wigner equation, SIAM J. Sci. Comput. 38 (2016), no. 3, A1483–A1507. 10.1137/16M105798XSearch in Google Scholar
[7] W. Wagner, A random cloud model for the Schrödinger equation, Kinet. Relat. Models 7 (2014), no. 2, 361–379. 10.3934/krm.2014.7.361Search in Google Scholar
[8] W. Wagner, A class of probabilistic models for the Schrödinger equation, Monte Carlo Methods Appl. 21 (2015), no. 2, 121–137. 10.1515/mcma-2014-0014Search in Google Scholar
[9] W. Wagner, A random cloud model for the Wigner equation, Kinet. Relat. Models 9 (2016), no. 1, 217–235. 10.3934/krm.2016.9.217Search in Google Scholar
[10] A. Zlotnik, The Numerov–Crank–Nicolson scheme on a non-uniform mesh for the time-dependent Schrödinger equation on the half-axis, Kinet. Relat. Models 8 (2015), no. 3, 587–613. 10.3934/krm.2015.8.587Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
