I am studying scattering theory right now in my QM class, and I'm attempting the Griffiths problem 11.4 as an exercise (it's not for homework). The problem is: Consider the case of low-energy scattering from a spherical delta-function shell: $V(r) = \alpha \delta(r-a)$. Calculate the scattering amplitude $f(\theta)$, the differential cross-section, and the total cross-section.

He outlines the following method in the section: in the exterior region where $V(r) = 0$, you get an expression for the wavefunction in terms of some complicated Hankel function expression $$ \psi(r,\theta) = A \sum i^l (2l+1)[j_l(kr) + ika_lh_l^{(1)}(kr) ] P_l(cos \theta). $$ You then match boundary conditions with the explicit solution to the wavefunction inside the region where $V(r) \neq 0$.

I'm confused how to carry this out. It seems that the solution of the wavefunction inside the region $r<a$ is your typical plane wave $f(r) = \sin(kr)/r$. Then I'm thinking that you should match $f(a) = \psi(a)$ and $\psi'(a)-f'(a) = - \frac{2m\alpha}{\hbar}^2 f(a)$. However, this doesn't appear to give me the right answer..

So is this the correct approach (if not, how do you go about these kinds of problems)?


The wavefunction inside the sphere can be expanded in terms of those bessel funcs. which are regular at the origin, say: $$ \psi(r,\theta)_{inside} = A \sum i^l (2l+1)[b_l j_l(kr) ] P_l(cos \theta). $$ The wavefunction inside the sphere need not be a plane wave, just can't blow up at the origin.

Since the potential is spherical, you can match boundary conditions $l$ mode by $l$ mode. For each $l$ you have two boundary conditions and two unknowns, $a_l$ and $b_l$, so you should be good to go.


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