{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:02:20Z","timestamp":1760238140704,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T00:00:00Z","timestamp":1660953600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Technology, Taiwan","award":["110-2112-M-003-006-MY3"],"award-info":[{"award-number":["110-2112-M-003-006-MY3"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the H-theorem in an isolated quantum harmonic oscillator through the time-dependent Schr\u00f6dinger equation. The effect of potential in producing entropy is investigated in detail, and we found that including a barrier potential into a harmonic trap would lead to the thermalization of the system, while a harmonic trap alone would not thermalize the system. During thermalization, Shannon entropy increases, which shows that a microscopic quantum system still obeys the macroscopic thermodynamics law. Meanwhile, initial coherent mechanical energy transforms to incoherent thermal energy during thermalization, which exhibiting the decoherence of an oscillating wave packet featured by a large decreasing of autocorrelation length. When reaching thermal equilibrium, the wave packet comes to a halt, with the density distributions both in position and momentum spaces well-fitted by a microcanonical ensemble of statistical mechanics.<\/jats:p>","DOI":"10.3390\/e24081163","type":"journal-article","created":{"date-parts":[[2022,8,21]],"date-time":"2022-08-21T22:23:13Z","timestamp":1661120593000},"page":"1163","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["H-Theorem in an Isolated Quantum Harmonic Oscillator"],"prefix":"10.3390","volume":"24","author":[{"given":"Che-Hsiu","family":"Hsueh","sequence":"first","affiliation":[{"name":"Department of Optoelectric Physics, Chinese Culture University, Taipei 111, Taiwan"},{"name":"Department of Physics, National Taiwan Normal University, Taipei 106, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6127-3115","authenticated-orcid":false,"given":"Chi-Ho","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Physics, National Changhua University of Education, Changhua 500, Taiwan"}]},{"given":"Tzyy-Leng","family":"Horng","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Feng Chia University, Taichung 407, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1682-2389","authenticated-orcid":false,"given":"Wen-Chin","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Physics, National Taiwan Normal University, Taipei 106, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1103\/RevModPhys.83.863","article-title":"Colloquium: Nonequilibrium dynamics of closed interacting quantum systems","volume":"83","author":"Polkovnikov","year":"2011","journal-title":"Rev. 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