Exact solutions of the Schrödinger equation for a class of hyperbolic potential well

doi:10.1088/1674-1056/ac3392

Abstract We propose a new scheme to study the exact solutions of a class of hyperbolic potential well. We first apply different forms of function transformation and variable substitution to transform the Schrödinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate. And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant, we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well. Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function. The linearly dependent relation between two eigenfunctions is also studied.

Keywords: hyperbolic potential well Schrödinger equation Wronskian determinant confluent Heun function

Received: 08 August 2021
Revised: 14 October 2021
Accepted manuscript online: 27 October 2021

PACS:	03.65.Ge	(Solutions of wave equations: bound states)
	03.65.Db	(Functional analytical methods)
	02.30.-f	(Function theory, analysis)
	02.60.Jh	(Numerical differentiation and integration)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11975196) and partially by SIP, Instituto Politecnico Nacional (IPN), Mexico (Grant No. 20210414). Prof. Dong started this work on Sabbatical Leave of IPN.

Corresponding Authors: Chang-Yuan Chen, Shi-Hai Dong
E-mail: chency@yctu.edu.cn;dongsh2@yahoo.com

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Xiao-Hua Wang(王晓华), Chang-Yuan Chen(陈昌远), Yuan You(尤源), Fa-Lin Lu(陆法林), Dong-Sheng Sun(孙东升), and Shi-Hai Dong(董世海) Exact solutions of the Schrödinger equation for a class of hyperbolic potential well 2022 Chin. Phys. B 31 040301

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