Determining a Scattering Cross Section (Quantum Mechanics)

[CDL]

Homework Statement

Consider scattering of a particle of mass $m$ on the potential
$$U(r) = \begin{cases}
0, & r \geq b\\
W, & r < b \\
\end{cases}$$
Where $W$ is some arbitrary chosen constant, and the radius $b$ is considered a small parameter. Find the cross section $\sigma$ in the limit $b \to 0$ (to avoid confusion, find the first non-zero term in the expansion of $\sigma$ over $b$). Prove in particular, that the limit does not depend on the energy $E$ of the particle.

Homework Equations

I am thinking of using partial wave analysis for this problem, since we have a "localised" potential. In particular, using

$$\sigma = \frac{4 \pi}{k^2} \sum_{l} (2l+1) \text{sin}^2 \delta_l$$
One thing that I am struggling to get my head around is how to find the phase shifts, $\delta_l$. After some searching around, I found the following formulas (not that I really understand them), $$\text{tan}\delta_l = \frac{k a j'_l(kb) - \beta_{l +} j_l(kb)}{kay'_{l}(ka) - \beta_{l+} y_l(ka)}$$ Where $j$ and $y$ denote the spherical Bessel functions. Also, $$\beta_{l+} = \frac{1}{\mathcal{R}} \frac{d\mathcal{R}}{dr} \bigg\rvert_{r = b}$$ with $$\mathcal{R} = e^{i \delta_l} \left(\text{cos}\delta_l \ j_l(kr) -\text{sin}\delta_l \ y_l(kr)\right)$$

The Attempt at a Solution

After using the equations in the above section, and 2 pages of working, I didn't get far. Those equation don't seem to use any information about the form of potential. I wanted to find the partial wave phase shifts until I hit the first non-zero one, and then sub it into the expansion for $\sigma$. I'm not sure how this would correspond to the first term in the expansion of $\sigma$ over $b$ though.

My whole understanding of this topic is kind of shaky, and reading Griffiths and the lecture notes doesn't seem to be getting me very far. Any suggestions on how to get a good grasp of this stuff?

 

[Gene Naden]

Well, I am still researching this but I thought I would get back to you with the one consideration I have found so far. That is, if $b<<1/k$ where $k$ is the incident wave vector then s wave scattering $(l=0)$ dominates. But then, maybe you already knew that.