{"title":"Analytical Solutions of the Schrödinger Equation for Two Confined Particles with the van der Waals Interaction","authors":"Ruijie Du","doi":"10.1007/s00601-024-01970-w","DOIUrl":null,"url":null,"abstract":"<div><p>We derive exact analytical solutions to the Schrödinger equation featuring a dual-scale potential, namely, a blend of a van der Waals (vdW) potential and an isotropic harmonic potential. The asymptotic behaviors of these solutions as <span>\\(r\\rightarrow 0\\)</span> and <span>\\(r\\rightarrow \\infty \\)</span> are also elucidated. These results are obtained through the approach we recently developed [arXiv: 2207.09377]. Using our results, we further calculate the <i>s</i>-wave and <i>p</i>-wave energy spectrums of two particles confined in an isotropic harmonic trap, with vdW inter-particle interaction. We compare our exact results and the ones given by the zero-range pseudopotential (ZRP) approaches, with either energy-dependent or energy-independent <i>s</i>-wave scattering length <span>\\(a_s\\)</span> or <i>p</i>-wave scattering volume <span>\\(V_p\\)</span>. It is shown that the results of ZRP approaches with energy-dependent <span>\\(a_s\\)</span> or <span>\\(V_p\\)</span> consist well with our exact ones, when the length scale <span>\\(\\beta _6\\)</span> of the vdW potential equals to or less than the length scale <span>\\(a_h\\)</span> of the confinement potential. Furthermore, when <span>\\(\\beta _6\\gg a_h\\)</span> (e.g., <span>\\(\\beta _6=10a_h\\)</span>) all the ZRP approaches fail. Our results are helpful for the research of confined ultracold atoms or molecules with strong vdW interactions.</p></div>","PeriodicalId":556,"journal":{"name":"Few-Body Systems","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Few-Body Systems","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00601-024-01970-w","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We derive exact analytical solutions to the Schrödinger equation featuring a dual-scale potential, namely, a blend of a van der Waals (vdW) potential and an isotropic harmonic potential. The asymptotic behaviors of these solutions as \(r\rightarrow 0\) and \(r\rightarrow \infty \) are also elucidated. These results are obtained through the approach we recently developed [arXiv: 2207.09377]. Using our results, we further calculate the s-wave and p-wave energy spectrums of two particles confined in an isotropic harmonic trap, with vdW inter-particle interaction. We compare our exact results and the ones given by the zero-range pseudopotential (ZRP) approaches, with either energy-dependent or energy-independent s-wave scattering length \(a_s\) or p-wave scattering volume \(V_p\). It is shown that the results of ZRP approaches with energy-dependent \(a_s\) or \(V_p\) consist well with our exact ones, when the length scale \(\beta _6\) of the vdW potential equals to or less than the length scale \(a_h\) of the confinement potential. Furthermore, when \(\beta _6\gg a_h\) (e.g., \(\beta _6=10a_h\)) all the ZRP approaches fail. Our results are helpful for the research of confined ultracold atoms or molecules with strong vdW interactions.
期刊介绍:
The journal Few-Body Systems presents original research work – experimental, theoretical and computational – investigating the behavior of any classical or quantum system consisting of a small number of well-defined constituent structures. The focus is on the research methods, properties, and results characteristic of few-body systems. Examples of few-body systems range from few-quark states, light nuclear and hadronic systems; few-electron atomic systems and small molecules; and specific systems in condensed matter and surface physics (such as quantum dots and highly correlated trapped systems), up to and including large-scale celestial structures.
Systems for which an equivalent one-body description is available or can be designed, and large systems for which specific many-body methods are needed are outside the scope of the journal.
The journal is devoted to the publication of all aspects of few-body systems research and applications. While concentrating on few-body systems well-suited to rigorous solutions, the journal also encourages interdisciplinary contributions that foster common approaches and insights, introduce and benchmark the use of novel tools (e.g. machine learning) and develop relevant applications (e.g. few-body aspects in quantum technologies).