Analytical Solutions of the Schrödinger Equation for Two Confined Particles with the van der Waals Interaction

IF 1.7 4区物理与天体物理 Q2 PHYSICS, MULTIDISCIPLINARY Few-Body Systems Pub Date : 2024-11-03 DOI:10.1007/s00601-024-01970-w

Ruijie Du

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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.

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具有范德华相互作用的两个密闭粒子的薛定谔方程的解析解

我们推导出了具有双尺度势（即范德华势和各向同性调和势的混合势）的薛定谔方程的精确解析解。我们还阐明了这些解作为 \(r\rightarrow 0\) 和 \(r\rightarrow \infty \) 的渐近行为。这些结果是通过我们最近开发的方法获得的[arXiv: 2207.09377]。利用我们的结果，我们进一步计算了约束在各向同性谐波阱中的两个粒子的s波和p波能谱，粒子间相互作用为vdW。我们比较了我们的精确结果和零射程伪势（ZRP）方法给出的结果，以及与能量相关或与能量无关的s波散射长度\(a_s\)或p波散射体积\(V_p\)。研究表明，当vdW势的长度尺度（\(\beta _6\）等于或小于约束势的长度尺度（\(a_h\）时，与能量相关的（\(a_s\)或\(V_p\)）ZRP方法的结果与我们的精确结果一致。此外，当\(\beta _6\gg a_h\)时（例如\(\beta _6=10a_h\) ），所有的ZRP方法都失败了。我们的结果有助于研究具有强vdW相互作用的约束超冷原子或分子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Few-Body Systems 物理-物理：综合

CiteScore

2.90

自引率

18.80%

发文量

审稿时长

6-12 weeks

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).