Delta Potential Born Approximation

Problem

Calculate the Born approximation to the differential and total cross sections for a particle of mass m off the $\delta$-function potential $V(r)=g\delta ^{3}(r)$.

Solution:

In Born approximation,

$f_{B}(\theta )=-{\frac {m}{2\pi \hbar ^{2}}}\int V(r')e^{-i\mathbf {q} \cdot \mathbf {r} '}d^{3}r'$

where $e^{-i\mathbf {q} \cdot \mathbf {r} '}$

i\hbar\frac{\partial \psi(\textbf{r},t)}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\textbf{r})\right]\psi(\textbf{r},t