A beam of particles of mass $m$ and energy $\hbar^{2} k^{2} / 2 m$ is incident on a target at the origin described by a spherically symmetric potential $V(r)$. Assuming the potential decays rapidly as $r \rightarrow \infty$, write down the asymptotic form of the wavefunction, defining the scattering amplitude $f(\theta)$.

Consider a free particle with energy $\hbar^{2} k^{2} / 2 m$. State, without proof, the general axisymmetric solution of the Schrödinger equation for $r>0$ in terms of spherical Bessel and Neumann functions $j_{\ell}$ and $n_{\ell}$, and Legendre polynomials $P_{\ell}(\ell=0,1,2, \ldots)$. Hence define the partial wave phase shifts $\delta_{\ell}$ for scattering from a potential $V(r)$ and derive the partial wave expansion for $f(\theta)$ in terms of phase shifts.

Now suppose

$$V(r)=\left\{\begin{array}{cc} \hbar^{2} \gamma^{2} / 2 m & r<a \\ 0 & r>a \end{array}\right.$$

with $\gamma>k$. Show that the S-wave phase shift $\delta_{0}$ obeys

$$\frac{\tanh (\kappa a)}{\kappa a}=\frac{\tan \left(k a+\delta_{0}\right)}{k a}$$

where $\kappa^{2}=\gamma^{2}-k^{2}$. Deduce that for an S-wave solution

$$f \rightarrow \frac{\tanh \gamma a-\gamma a}{\gamma} \quad \text { as } \quad k \rightarrow 0$$

[You may assume : $\quad \exp (i k r \cos \theta)=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(k r) P_{\ell}(\cos \theta)$

and $j_{\ell}(\rho) \sim \frac{1}{\rho} \sin (\rho-\ell \pi / 2), \quad n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos (\rho-\ell \pi / 2) \quad$ as $\left.\quad \rho \rightarrow \infty .\right]$