A beam of particles is incident on a central potential $V(r)(r=|\mathbf{x}|)$ that vanishes for $r>R$. Define the differential cross-section $d \sigma / d \Omega$.

Given that each incoming particle has momentum $\hbar \mathbf{k}$, explain the relevance of solutions to the time-independent Schrödinger equation with the asymptotic form

$$\psi(\mathbf{x}) \sim e^{i \mathbf{k} \cdot \mathbf{x}}+f(\hat{\mathbf{x}}) \frac{e^{i k r}}{r}$$

as $r \rightarrow \infty$, where $k=|\mathbf{k}|$ and $\hat{\mathbf{x}}=\mathbf{x} / r$. Write down a formula that determines $d \sigma / d \Omega$ in this case.

Write down the time-independent Schrödinger equation for a particle of mass $m$ and energy $E=\frac{\hbar^{2} k^{2}}{2 m}$ in a central potential $V(r)$, and show that it allows a solution of the form

$$\psi(\mathbf{x})=e^{i \mathbf{k} \cdot \mathbf{x}}-\frac{m}{2 \pi \hbar^{2}} \int d^{3} x^{\prime} \frac{e^{i k\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} V\left(r^{\prime}\right) \psi\left(\mathbf{x}^{\prime}\right) .$$

Show that this is consistent with $(*)$ and deduce an expression for $f(\hat{\mathbf{x}})$. Obtain the Born approximation for $f(\hat{\mathbf{x}})$, and show that $f(\hat{\mathbf{x}})=F(k \hat{\mathbf{x}}-\mathbf{k})$, where

$$F(\mathbf{q})=-\frac{m}{2 \pi \hbar^{2}} \int d^{3} x e^{-i \mathbf{q} \cdot \mathbf{x}} V(r)$$

Under what conditions is the Born approximation valid?

Obtain a formula for $f(\hat{\mathbf{x}})$ in terms of the scattering angle $\theta$ in the case that

$$V(r)=K \frac{e^{-\mu r}}{r},$$

for constants $K$ and $\mu$. Hence show that $f(\hat{\mathbf{x}})$ is independent of $\hbar$ in the limit $\mu \rightarrow 0$, when expressed in terms of $\theta$ and the energy $E$.

[You may assume that $\left.\left(\nabla^{2}+k^{2}\right)\left(\frac{e^{i k r}}{r}\right)=-4 \pi \delta^{3}(\mathbf{x}) .\right]$