A beam of particles of mass $m$ and momentum $p=\hbar k$ is incident along the $z$-axis. The beam scatters off a spherically symmetric potential $V(r)$. Write down the asymptotic form of the wavefunction in terms of the scattering amplitude $f(\theta)$.

The incoming plane wave and the scattering amplitude can be expanded in partial waves as,

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

where $P_{l}$ are Legendre polynomials. Define the $S$-matrix. Assuming that the S-matrix is unitary, explain why we can write

$$f_{l}=e^{i \delta_{l}} \sin \delta_{l}$$

for some real phase shifts $\delta_{l}$. Obtain an expression for the total cross-section $\sigma_{T}$ in terms of the phase shifts $\delta_{l}$.

[Hint: You may use the orthogonality of Legendre polynomials:

$$\left.\int_{-1}^{+1} d w P_{l}(w) P_{l^{\prime}}(w)=\frac{2}{2 l+1} \delta_{l l^{\prime}} .\right]$$

Consider the repulsive, spherical potential

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

where $V_{0}=\hbar^{2} \gamma^{2} / 2 m$. By considering the s-wave solution to the Schrödinger equation, show that

$$\frac{\tan \left(k a+\delta_{0}\right)}{k a}=\frac{\tanh \left(\sqrt{\gamma^{2}-k^{2}} a\right)}{\sqrt{\gamma^{2}-k^{2}} a}$$

For low momenta, $k a \ll 1$, compute the s-wave contribution to the total cross-section. Comment on the physical interpretation of your result in the limit $\gamma a \rightarrow \infty$.