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