Principles of Quantum Mechanics · Problem 19.1

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · scattering-theory-classical

Problem

Rutherford scattering. A particle of charge

q_1

and mass

m

scatters off a fixed Coulomb center of charge

q_2

. The potential energy is

U(r) = \frac{q_1 q_2}{r}

(in Gaussian units). Show that the differential cross section is given by the Rutherford formula:

\frac{d\sigma}{d\Omega} = \left(\frac{q_1 q_2}{4E}\right)^2 \frac{1}{\sin^4(\theta/2)}

where

E = \frac{1}{2}mv^2

is the kinetic energy of the incoming particle.