On an Exactly Solvable Two-Body Problem in Two-Dimensional Quantum Mechanics

IF 1.7 4区物理与天体物理 Q2 PHYSICS, MULTIDISCIPLINARY Few-Body Systems Pub Date : 2023-10-03 DOI:10.1007/s00601-023-01859-0

Roman Ya. Kezerashvili, Jianning Luo, Claudio R. Malvino

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引用次数: 2

Abstract

It is well known that exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two-dimensional (2D) problem for two particles enclosed in a circle, confined in an oscillatory well, trapped in a magnetic field, interacting via the Coulomb, Kratzer, and modified Kratzer potentials. In the framework of the Nikiforov–Uvarov method, we transform 2D Schrödinger equations with potentials for which the three-dimensional Schrödinger equation is exactly solvable, into a second-order differential equation of a hypergeometric-type via transformations of coordinates and particular substitutions. Within this unified approach which also has pedagogical merit, we obtain exact analytical solutions for wave functions in terms of special functions such as a hypergeometric function, confluent hypergeometric function, and solutions of Kummer’s, Laguerre’s, and Bessel’s differential equations. We present the energy spectrum for any arbitrary state with the azimuthal number m. Interesting aspects of the solutions unique to the 2D case are discussed.

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关于二维量子力学中一个可精确求解的二体问题

众所周知，精确可解模型在量子物理的许多领域中发挥着极其重要的作用。在这项研究中，Schrödinger方程被应用于二维（2D）问题的求解，这两个粒子被封闭在一个圆中，被限制在振荡阱中，被困在磁场中，通过库仑、Kratzer和修正的Kratzer势相互作用。在Nikiforov–Uvarov方法的框架下，我们通过坐标变换和特定置换，将三维薛定谔方程完全可解的具有势的二维薛定谔方程转化为超几何类型的二阶微分方程。在这种同样具有教学价值的统一方法中，我们获得了波函数在特殊函数方面的精确解析解，如超几何函数、合流超几何函数以及Kummer、Laguerre和Bessel微分方程的解。我们给出了方位角为m的任意状态的能谱。讨论了二维情况下唯一解的有趣方面。

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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).