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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)?

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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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