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
For a delta function potential
where is a constant representing the strength of the delta function, so we get
For low energy, so as well, so , and we get
Our earlier result using partial wave analysis is
This gives a differential cross section and total cross section of
The low energy result 5 from the Born approximation is, in terms of :
so it agrees with the partial wave result if . This is equivalent to the condition that , 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.