量子力学中能量本征值问题的拉普拉斯方法

Q2 Physics and Astronomy Quantum Reports Pub Date : 2022-08-15 DOI:10.3390/quantum5020024
J. Canfield, A. Galler, J. Freericks
{"title":"量子力学中能量本征值问题的拉普拉斯方法","authors":"J. Canfield, A. Galler, J. Freericks","doi":"10.3390/quantum5020024","DOIUrl":null,"url":null,"abstract":"Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.","PeriodicalId":34124,"journal":{"name":"Quantum Reports","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics\",\"authors\":\"J. Canfield, A. Galler, J. Freericks\",\"doi\":\"10.3390/quantum5020024\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.\",\"PeriodicalId\":34124,\"journal\":{\"name\":\"Quantum Reports\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-08-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quantum Reports\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/quantum5020024\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Physics and Astronomy\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Reports","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/quantum5020024","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0

摘要

量子力学大约有十几个完全可解的势。通常,它们的时间无关的薛定谔方程是通过使用束缚态的广义级数解(使用Fröbenius方法)和连续态的解析延拓(如果存在)来求解的。在这项工作中,我们提出了一种解决这些问题的替代方法,基于拉普拉斯方法。该技术对束缚态和连续态使用了类似的过程。它最初是薛定谔在求解氢的波函数时使用的。狄拉克也提倡使用这种方法。我们讨论了为什么它是一种强大的方法来解决所有波函数用合流超几何函数表示的问题,特别是对于连续体解,它可以通过易于编程的轮廓积分来确定。
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
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
The Laplace Method for Energy Eigenvalue Problems in Quantum Mechanics
Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.
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
Quantum Reports
Quantum Reports Physics and Astronomy-Physics and Astronomy (miscellaneous)
CiteScore
3.30
自引率
0.00%
发文量
33
审稿时长
10 weeks
期刊最新文献
Nitrogen-Related High-Spin Vacancy Defects in Bulk (SiC) and 2D (hBN) Crystals: Comparative Magnetic Resonance (EPR and ENDOR) Study Fisher Information for a System Composed of a Combination of Similar Potential Models A Normalization Condition for the Probability Current in Some Remarkable Cases The Many-Worlds Interpretation of Quantum Mechanics: Current Status and Relation to Other Interpretations Tomographic Universality of the Discrete Wigner Function
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1