Principles of Quantum Mechanics · Problem 19.20

Scattering Theory — R. Shankar

Physics — Quantum Mechanics · born-approximation

Problem

The Born approximation gives the scattering amplitude as

f^{(1)}(\theta) = -\frac{m}{2\pi\hbar^2}\tilde{V}(\mathbf{q})

where

\mathbf{q} = \mathbf{k}' - \mathbf{k}

and

\tilde{V}

is the Fourier transform of

V

. (a) For the potential

V(r) = V_0 e^{-r/a}/r

, compute

\tilde{V}(q)

. (b) Write down the differential cross section in the Born approximation. (c) Integrate to find the total cross section.