大学物理 ›› 2023, Vol. 42 ›› Issue (6): 21-.doi: 10.16854/j.cnki.1000-0712.220322

• 教学讨论 • 上一篇    下一篇

一维 th(λx)势散射的研究

杨维   

  1. 桂林理工大学 理学院,广西  桂林100875 
  • 收稿日期:2022-07-01 修回日期:2022-08-04 出版日期:2023-07-01 发布日期:2023-07-06
  • 作者简介:杨维(1987—),男,湖北洪湖人,桂林理工大学理学院讲师,博士,主要从事数学物理方法教学和散射理论的研究工作.Email:weiyang@glut.edu.cn
  • 基金资助:
    广西科学计划基金(2020AC20014);校级科研基金(GUTQDJJ2019206)资助

One-dimensional potential scattering of th(λx)

YANG Wei   

  1. College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
  • Received:2022-07-01 Revised:2022-08-04 Online:2023-07-01 Published:2023-07-06

摘要: 量子力学中的散射问题一直是学习的难点,也是理论研究的基础. 本文是为了理解散射提供一个精确可解的例子. 我们获得了一维定态薛定谔方程在 th(λx) 势下的精确解,并在精确解的基础上,利用超几何函数的渐近展开性质,计算出相应的反射系数和透射系数. 同时还发现当常数 λ 趋于无穷大极限时,th(λx) 势与阶梯势对应的反射系数和透射系数正好相等.

关键词: 散射理论, 超几何方程, 渐近展开, 精确解

Abstract:  The scattering problem in quantum mechanics has always been a difficult point of study and the basis of theoretical research. The purpose of this paper is to provide an exact solvable example for understanding scattering. We obtain the exact solution of one-dimensional time-independent Schrodinger equation with th(λx) potential, basis on this exact solution, and using the asymptotic expansion property of hypergeometric function, the corresponding reflection and transmission coefficients are calculated. At the same time, it is also found that when the constant λ approaching to infinite limit, the reflection and transmission coefficients of th(λx) potential and the step function potential are exactly equal to each other.

Key words:  scattering theory, hypergeometric equation, asymptotic expansion, exact solution