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

doi:10.1088/1674-1056/ac3392

中国物理B ›› 2022, Vol. 31 ›› Issue (4): 40301-040301.doi: 10.1088/1674-1056/ac3392

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

Xiao-Hua Wang(王晓华)¹, Chang-Yuan Chen(陈昌远)^1,†, Yuan You(尤源)¹, Fa-Lin Lu(陆法林)¹, Dong-Sheng Sun(孙东升)¹, and Shi-Hai Dong(董世海)^2,3,‡

1 School of Physics and Electronic Engineering, Yancheng Teachers University, Yancheng 224007, China;
2 Research Center for Quantum Physics, Huzhou University, Huzhou 313000, China;
3 Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, UPALM, C. P 07700, CDMX, Mexico

收稿日期:2021-08-08 修回日期:2021-10-14 接受日期:2021-10-27 出版日期:2022-03-16 发布日期:2022-03-25
通讯作者: Chang-Yuan Chen, Shi-Hai Dong E-mail:chency@yctu.edu.cn;dongsh2@yahoo.com
基金资助:
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.

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

Xiao-Hua Wang(王晓华)¹, Chang-Yuan Chen(陈昌远)^1,†, Yuan You(尤源)¹, Fa-Lin Lu(陆法林)¹, Dong-Sheng Sun(孙东升)¹, and Shi-Hai Dong(董世海)^2,3,‡

1 School of Physics and Electronic Engineering, Yancheng Teachers University, Yancheng 224007, China;
2 Research Center for Quantum Physics, Huzhou University, Huzhou 313000, China;
3 Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, UPALM, C. P 07700, CDMX, Mexico

Received:2021-08-08 Revised:2021-10-14 Accepted:2021-10-27 Online:2022-03-16 Published:2022-03-25
Contact: Chang-Yuan Chen, Shi-Hai Dong E-mail:chency@yctu.edu.cn;dongsh2@yahoo.com
Supported by:
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.

摘要/Abstract

摘要： 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.

关键词: hyperbolic potential well, Schrödinger equation, Wronskian determinant, confluent Heun function

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.

Key words: hyperbolic potential well, Schrödinger equation, Wronskian determinant, confluent Heun function

中图分类号: (Solutions of wave equations: bound states)

03.65.Ge

03.65.Db (Functional analytical methods) 02.30.-f (Function theory, analysis) 02.60.Jh (Numerical differentiation and integration)

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[J]. 中国物理B, 2022, 31(4): 40301-040301.

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

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

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