A beam of particles of mass $m$ and momentum $p=\hbar k$, incident along the $z$-axis, is scattered by a spherically symmetric potential $V(r)$, where $V(r)=0$ for large $r$. State the boundary conditions on the wavefunction as $r \rightarrow \infty$ and hence define the scattering amplitude $f(\theta)$, where $\theta$ is the scattering angle.

Given that, for large $r$,

$$e^{i k r \cos \theta}=\frac{1}{2 i k r} \sum_{l=0}^{\infty}(2 l+1)\left(e^{i k r}-(-1)^{l} e^{-i k r}\right) P_{l}(\cos \theta)$$

explain how the partial-wave expansion can be used to define the phase shifts $\delta_{l}(k)(l=$ $0,1,2, \ldots)$. Furthermore, given that $d \sigma / d \Omega=|f(\theta)|^{2}$, derive expressions for $f(\theta)$ and the total cross-section $\sigma$ in terms of the $\delta_{l}$.

In a particular case $V(r)$ is given by

$$V(r)=\left\{\begin{array}{cl} \infty, & r<a \\ -V_{0}, & a<r<2 a \\ 0, & r>2 a \end{array}\right.$$

where $V_{0}>0$. Show that the $\mathrm{S}$-wave phase shift $\delta_{0}$ satisfies

$$\tan \left(\delta_{0}\right)=\frac{k \cos (2 k a)-\kappa \cot (\kappa a) \sin (2 k a)}{k \sin (2 k a)+\kappa \cot (\kappa a) \cos (2 k a)},$$

where $\kappa^{2}=2 m V_{0} / \hbar^{2}+k^{2}$.

Derive an expression for the scattering length $a_{s}$ in terms of $\kappa$. Find the values of $\kappa$ for which $\left|a_{s}\right|$ diverges and briefly explain their physical significance.