(a) The $s$-wave solution $\psi_{0}$ for the scattering problem of a particle of mass $m$ and momentum $\hbar k$ has the asymptotic form

$$\psi_{0}(r) \sim \frac{A}{r}[\sin (k r)+g(k) \cos (k r)]$$

Define the phase shift $\delta_{0}$ and verify that $\tan \delta_{0}=g(k)$.

(b) Define the scattering amplitude $f$. For a spherically symmetric potential of finite range, starting from $\sigma_{T}=\int|f|^{2} d \Omega$, derive the expression

$$\sigma_{T}=\frac{4 \pi}{k^{2}} \sum_{l=0}^{\infty}(2 l+1) \sin ^{2} \delta_{l}$$

giving the cross-section $\sigma_{T}$ in terms of the phase shifts $\delta_{l}$ of the partial waves.

(c) For $g(k)=-k / K$ with $K>0$, show that a bound state exists and compute its energy. Neglecting the contributions from partial waves with $l>0$, show that

$$\sigma_{T} \approx \frac{4 \pi}{K^{2}+k^{2}}$$

(d) For $g(k)=\gamma /\left(K_{0}-k\right)$ with $K_{0}>0, \gamma>0$ compute the $s$-wave contribution to $\sigma_{T}$. Working to leading order in $\gamma \ll K_{0}$, show that $\sigma_{T}$ has a local maximum at $k=K_{0}$. Interpret this fact in terms of a resonance and compute its energy and decay width.