一维th(λx)势散射的研究

One-dimensional potential scattering of th(λx)

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.

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 constantλapproaching to infinite limit,the reflection and transmission coefficients of th(λx)potential and the step function potential are exactly equal to each other.