半经典区域中的量子动力学计算

IF 16.3 1区数学 Q1 MATHEMATICS

Acta Numerica Pub Date : 2020-02-01 DOI:10.1017/S0962492920000033

C. Lasser, C. Lubich

{"title":"半经典区域中的量子动力学计算","authors":"C. Lasser, C. Lubich","doi":"10.1017/S0962492920000033","DOIUrl":null,"url":null,"abstract":"The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn’s semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.","PeriodicalId":48863,"journal":{"name":"Acta Numerica","volume":"29 1","pages":"229 - 401"},"PeriodicalIF":16.3000,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0962492920000033","citationCount":"58","resultStr":"{\"title\":\"Computing quantum dynamics in the semiclassical regime\",\"authors\":\"C. Lasser, C. Lubich\",\"doi\":\"10.1017/S0962492920000033\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn’s semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.\",\"PeriodicalId\":48863,\"journal\":{\"name\":\"Acta Numerica\",\"volume\":\"29 1\",\"pages\":\"229 - 401\"},\"PeriodicalIF\":16.3000,\"publicationDate\":\"2020-02-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1017/S0962492920000033\",\"citationCount\":\"58\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Numerica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/S0962492920000033\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Numerica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0962492920000033","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 58

摘要

半经典尺度的含时多粒子薛定谔方程描述了分子中原子核的量子动力学。它带来了高振荡和高维的综合计算挑战。本文综述和研究了对小半经典参数具有鲁棒性的数值方法。我们提出并分析了变化演化的高斯波包、Hagedorn的半经典波包、解冻和冻结高斯的连续叠加，以及直接计算可观测值期望值的Wigner函数方法。对于所有这些方法来说，充分利用经典力学是至关重要的。还讨论了时间积分和高维求积的产生方面。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Computing quantum dynamics in the semiclassical regime

The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn’s semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.