{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:50:45Z","timestamp":1747198245398,"version":"3.40.5"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>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 Schr\u00f6dinger 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 Schr\u00f6dinger 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.<\/jats:p>","DOI":"10.1515\/mcma-2017-0118","type":"journal-article","created":{"date-parts":[[2017,11,4]],"date-time":"2017-11-04T22:15:59Z","timestamp":1509833759000},"page":"221-240","source":"Crossref","is-referenced-by-count":0,"title":["A random cloud algorithm for the Schr\u00f6dinger equation"],"prefix":"10.1515","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4293-8924","authenticated-orcid":false,"given":"Markus","family":"Kraft","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering and Biotechnology , University of Cambridge , Philippa Fawcett Drive , Cambridge CB3 0AS , United Kingdom"}]},{"given":"Wolfgang","family":"Wagner","sequence":"additional","affiliation":[{"name":"Weierstrass Institute for Applied Analysis and Stochastics , Mohrenstr. 39, 10117 Berlin , Germany"}]}],"member":"374","published-online":{"date-parts":[[2017,11,4]]},"reference":[{"key":"2023040101345242243_j_mcma-2017-0118_ref_001_w2aab3b7b5b1b6b1ab1b5b1Aa","unstructured":"R.  Becerril, F. S.  Guzm\u00e1n, A.  Rend\u00f3n-Romero and S.  Valdez-Alvarado,\nSolving the time-dependent Schr\u00f6dinger equation using finite difference methods,\nRev. Mex. Fis. E 54 (2008), 120\u2013132."},{"key":"2023040101345242243_j_mcma-2017-0118_ref_002_w2aab3b7b5b1b6b1ab1b5b2Aa","unstructured":"P.  Br\u00e9maud,\nMarkov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,\nTexts Appl. Math. 31,\nSpringer, New York, 2013."},{"key":"2023040101345242243_j_mcma-2017-0118_ref_003_w2aab3b7b5b1b6b1ab1b5b3Aa","unstructured":"D. O.  Hayward,\nQuantum Mechanics for Chemists,\nTutor. Chem. Texts 14,\nThe Royal Society of Chemistry, London, 2002."},{"key":"2023040101345242243_j_mcma-2017-0118_ref_004_w2aab3b7b5b1b6b1ab1b5b4Aa","doi-asserted-by":"crossref","unstructured":"M.  Kraft and W.  Wagner,\nAn improved stochastic algorithm for temperature-dependent homogeneous gas phase reactions,\nJ. Comput. Phys. 185 (2003), 139\u2013157.\n10.1016\/S0021-9991(02)00051-7","DOI":"10.1016\/S0021-9991(02)00051-7"},{"key":"2023040101345242243_j_mcma-2017-0118_ref_005_w2aab3b7b5b1b6b1ab1b5b5Aa","doi-asserted-by":"crossref","unstructured":"M.  Kraft and W.  Wagner,\nNumerical study of a stochastic particle method for homogeneous gas-phase reactions,\nComput. Math. Appl. 45 (2003), no. 1\u20133, 329\u2013349.\n10.1016\/S0898-1221(03)80022-6","DOI":"10.1016\/S0898-1221(03)80022-6"},{"key":"2023040101345242243_j_mcma-2017-0118_ref_006_w2aab3b7b5b1b6b1ab1b5b6Aa","doi-asserted-by":"crossref","unstructured":"O.  Muscato and W.  Wagner,\nA class of stochastic algorithms for the Wigner equation,\nSIAM J. Sci. Comput. 38 (2016), no. 3, A1483\u2013A1507.\n10.1137\/16M105798X","DOI":"10.1137\/16M105798X"},{"key":"2023040101345242243_j_mcma-2017-0118_ref_007_w2aab3b7b5b1b6b1ab1b5b7Aa","doi-asserted-by":"crossref","unstructured":"W.  Wagner,\nA random cloud model for the Schr\u00f6dinger equation,\nKinet. Relat. Models 7 (2014), no. 2, 361\u2013379.\n10.3934\/krm.2014.7.361","DOI":"10.3934\/krm.2014.7.361"},{"key":"2023040101345242243_j_mcma-2017-0118_ref_008_w2aab3b7b5b1b6b1ab1b5b8Aa","doi-asserted-by":"crossref","unstructured":"W.  Wagner,\nA class of probabilistic models for the Schr\u00f6dinger equation,\nMonte Carlo Methods Appl. 21 (2015), no. 2, 121\u2013137.","DOI":"10.1515\/mcma-2014-0014"},{"key":"2023040101345242243_j_mcma-2017-0118_ref_009_w2aab3b7b5b1b6b1ab1b5b9Aa","doi-asserted-by":"crossref","unstructured":"W.  Wagner,\nA random cloud model for the Wigner equation,\nKinet. Relat. Models 9 (2016), no. 1, 217\u2013235.","DOI":"10.3934\/krm.2016.9.217"},{"key":"2023040101345242243_j_mcma-2017-0118_ref_010_w2aab3b7b5b1b6b1ab1b5c10Aa","doi-asserted-by":"crossref","unstructured":"A.  Zlotnik,\nThe Numerov\u2013Crank\u2013Nicolson scheme on a non-uniform mesh for the time-dependent Schr\u00f6dinger equation on the half-axis,\nKinet. Relat. Models 8 (2015), no. 3, 587\u2013613.\n10.3934\/krm.2015.8.587","DOI":"10.3934\/krm.2015.8.587"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.degruyter.com\/view\/j\/mcma.2017.23.issue-4\/mcma-2017-0118\/mcma-2017-0118.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2017-0118\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2017-0118\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T20:49:09Z","timestamp":1680382149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2017-0118\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,4]]},"references-count":10,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,12,1]]},"published-print":{"date-parts":[[2017,12,1]]}},"alternative-id":["10.1515\/mcma-2017-0118"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2017-0118","relation":{},"ISSN":["1569-3961","0929-9629"],"issn-type":[{"type":"electronic","value":"1569-3961"},{"type":"print","value":"0929-9629"}],"subject":[],"published":{"date-parts":[[2017,11,4]]}}}