A steady beam of particles, having wavenumber $k$ and moving in the $z$ direction, scatters on a spherically-symmetric potential. Write down the asymptotic form of the wave function at large $r$.

The incoming wave is written as a partial-wave series

$$\sum_{\ell=0}^{\infty} \chi_{\ell}(k r) P_{\ell}(\cos \theta)$$

Show that for large $r$

$$\chi_{\ell}(k r) \sim \frac{\ell+\frac{1}{2}}{i k r}\left(e^{i k r}-(-1)^{\ell} e^{-i k r}\right)$$

and calculate $\chi_{0}(k r)$ and $\chi_{1}(k r)$ for all $r$.

Write down the second-order differential equation satisfied by the $\chi_{\ell}(k r)$. Construct a second linearly-independent solution for each $\ell$ that is singular at $r=0$ and, when it is suitably normalised, has large- $r$ behaviour

$$\frac{\ell+\frac{1}{2}}{i k r}\left(e^{i k r}+(-1)^{\ell} e^{-i k r}\right)$$