# Born approximation for a spherical delta function shell

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 11.13.

Earlier we looked at scattering from a delta function spherical shell for a low energy incident particle, using partial wave analysis. This was a fairly complex task, as it involved matching interior and exterior wave functions at the delta function boundary.

Here, we’ll calculate the scattering amplitude using the first Born approximation. For a spherically symmetric potential, the approximation is

$\displaystyle f\left(\theta\right)\approx-\frac{2m}{\hbar^{2}\kappa}\int_{0}^{\infty}V\left(r\right)r\sin\left(\kappa r\right)dr \ \ \ \ \ (1)$

where

$\displaystyle \kappa\equiv2k\sin\frac{\theta}{2} \ \ \ \ \ (2)$

For a delta function potential

$\displaystyle V\left(r\right)=\alpha\delta\left(r-a\right) \ \ \ \ \ (3)$

where ${\alpha}$ is a constant representing the strength of the delta function, so we get

$\displaystyle f\left(\theta\right)\approx-\frac{2m\alpha a}{\hbar^{2}\kappa}\sin\left(\kappa a\right) \ \ \ \ \ (4)$

For low energy, ${ka\ll1}$ so ${\kappa a\ll1}$ as well, so ${\sin\left(\kappa a\right)\approx\kappa a}$, and we get

$\displaystyle f\left(\theta\right)\approx-\frac{2m\alpha a^{2}}{\hbar^{2}} \ \ \ \ \ (5)$

Our earlier result using partial wave analysis is

$\displaystyle f\left(\theta\right)\approx-\frac{\beta a}{1+\beta} \ \ \ \ \ (6)$

where

$\displaystyle \beta\equiv\frac{2m\alpha a}{\hbar^{2}} \ \ \ \ \ (7)$

This gives a differential cross section and total cross section of

 $\displaystyle \frac{d\sigma}{d\Omega}$ $\displaystyle =$ $\displaystyle \left|f\left(\theta\right)\right|^{2}=\beta^{2}a^{2}\ \ \ \ \ (8)$ $\displaystyle \sigma$ $\displaystyle =$ $\displaystyle 4\pi\beta^{2}a^{2} \ \ \ \ \ (9)$

The low energy result 5 from the Born approximation is, in terms of ${\beta}$:

$\displaystyle f\left(\theta\right)\approx-\beta a \ \ \ \ \ (10)$

so it agrees with the partial wave result if ${\beta\ll1}$. This is equivalent to the condition that ${\alpha\ll\hbar^{2}/2ma}$, in other words, that the potential is weak. This was the main assumption in deriving the Born approximation, so in this limit, the results are consistent.