The wave function for a single particle with a potential $V(r)$ has the asymptotic form for large $r$

$$\psi(r, \theta) \sim e^{i k r \cos \theta}+f(\theta) \frac{e^{i k r}}{r} .$$

How is $f(\theta)$ related to observable quantities? Show how $f(\theta)$ can be expressed in terms of phase shifts $\delta_{\ell}(k)$ for $\ell=0,1,2, \ldots$..

Assume that $V(r)=0$ for $r \geq a$, and let $R_{\ell}(r)$ denote the solution of the radial Schrödinger equation, regular at $r=0$, with energy $\hbar^{2} k^{2} / 2 m$ and angular momentum $\ell$. Let $N_{\ell}(k)=a R_{\ell}^{\prime}(a) / R_{\ell}(a)$. Show that

$$\tan \delta_{\ell}(k)=\frac{N_{\ell}(k) j_{\ell}(k a)-k a j_{\ell}^{\prime}(k a)}{N_{\ell}(k) n_{\ell}(k a)-k a n_{\ell}^{\prime}(k a)} .$$

Assuming that $N_{\ell}(k)$ is a smooth function for $k \approx 0$, determine the expected behaviour of $\delta_{\ell}(k)$ as $k \rightarrow 0$. Show that for $k \rightarrow 0$ then $f(\theta) \rightarrow c$, with $c$ a constant, and determine $c$ in terms of $N_{0}(0)$.

[For $V=0$ the two independent solutions of the radial Schrödinger equation are $j_{\ell}(k r)$ and $n_{\ell}(k r)$ with

$$\begin{aligned} j_{\ell}(\rho) & \sim \frac{1}{\rho} \sin \left(\rho-\frac{1}{2} \ell \pi\right), n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos \left(\rho-\frac{1}{2} \ell \pi\right) \text { as } \rho \rightarrow \\ j_{\ell}(\rho) & \propto \rho^{\ell}, n_{\ell}(\rho) \propto \rho^{-\ell-1} \text { as } \rho \rightarrow 0 \\ e^{i \rho \cos \theta} &=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(\rho) P_{\ell}(\cos \theta), \\ j_{0}(\rho) &=\frac{\sin \rho}{\rho}, \quad n_{0}(\rho)=-\frac{\cos \rho}{\rho} \end{aligned}$$