Principles of Quantum Mechanics · Problem 19.4

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · born-approximation

Problem

Use the Born approximation to calculate

d\sigma/d\Omega

for scattering from a spherical well:

V(r) = \begin{cases} -V_0, & r < a \\ 0, & r > a \end{cases}

Show that

\frac{d\sigma}{d\Omega} = \left(\frac{2mV_0 a^3}{\hbar^2}\right)^2 \left[\frac{\sin(\kappa a) - \kappa a \cos(\kappa a)}{(\kappa a)^3}\right]^2

where

\kappa = 2k\sin(\theta/2)

. What is the total cross section in the low-energy limit (

ka \ll 1