Daily Archives: Wed, 17 June 2015

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.