A beam of particles of mass $m$ and momentum $p=\hbar k$ is incident along the $z$-axis. Write down the asymptotic form of the wave function which describes scattering under the influence of a spherically symmetric potential $V(r)$ and which defines the scattering amplitude $f(\theta)$.

Given that, for large $r$,

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

show how to derive the partial-wave expansion of the scattering amplitude in the form

$$f(\theta)=\frac{1}{k} \sum_{l=0}^{\infty}(2 l+1) e^{i \delta_{l}} \sin \delta_{l} P_{l}(\cos \theta)$$

Obtain an expression for the total cross-section, $\sigma$.

Let $V(r)$ have the form

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

where $V_{0}=\frac{\hbar^{2}}{2 m} \gamma^{2}$

Show that the $l=0$ phase-shift $\delta_{0}$ satisfies

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

where $\kappa^{2}=k^{2}+\gamma^{2}$.

Assume $\gamma$ to be large compared with $k$ so that $\kappa$ may be approximated by $\gamma$. Show, using graphical methods or otherwise, that there are values for $k$ for which $\delta_{0}(k)=n \pi$ for some integer $n$, which should not be calculated. Show that the smallest value, $k_{0}$, of $k$ for which this condition holds certainly satisfies $k_{0}<3 \pi / 2 a$.