(a) For the quantum scattering of a beam of particles in three dimensions off a spherically symmetric potential $V(r)$ that vanishes at large $r$, discuss the boundary conditions satisfied by the wavefunction $\psi$ and define the scattering amplitude $f(\theta)$. Assuming the asymptotic form

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

state the constraints on $f_{l}$ imposed by the unitarity of the $S$-matrix and define the phase shifts $\delta_{l}$.

(b) For $V_{0}>0$, consider the specific potential

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

(i) Show that the s-wave phase shift $\delta_{0}$ obeys

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

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

(ii) Compute the scattering length $a_{s}$ and find for which values of $\kappa$ it diverges. Discuss briefly the physical interpretation of the divergences. [Hint: you may find this trigonometric identity useful

$$\left.\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} .\right]$$