Principles of Quantum Mechanics · Problem 19.3

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · born-approximation

Problem

Calculate the Born approximation to the differential cross section

d\sigma/d\Omega

for scattering from a Yukawa potential:

V(r) = \beta \frac{e^{-\mu r}}{r}

where

\beta

and

\mu

are constants. Show that

\frac{d\sigma}{d\Omega} = \left(\frac{2m\beta}{\hbar^2}\right)^2 \frac{1}{(\mu^2 + \kappa^2)^2}

where

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

. By taking

\mu \to 0

with

\beta = q_1 q_2

, recover the Rutherford formula.