A beam of particles each of mass $m$ and energy $\hbar^{2} k^{2} /(2 m)$ scatters off an axisymmetric potential $V$. In the first Born approximation the scattering amplitude is

$$f(\theta)=-\frac{m}{2 \pi \hbar^{2}} \int e^{-i\left(\mathbf{k}-\mathbf{k}_{0}\right) \cdot \mathbf{x}^{\prime}} V\left(\mathbf{x}^{\prime}\right) d^{3} x^{\prime}$$

where $\mathbf{k}_{0}=(0,0, k)$ is the wave vector of the incident particles and $\mathbf{k}=(k \sin \theta, 0, k \cos \theta)$ is the wave vector of the outgoing particles at scattering angle $\theta$ (and $\phi=0$ ). Let $\mathbf{q}=\mathbf{k}-\mathbf{k}_{0}$ and $q=|\mathbf{q}|$. Show that when the scattering potential $V$ is spherically symmetric the expression $(*)$ simplifies to

$$f(\theta)=-\frac{2 m}{\hbar^{2} q} \int_{0}^{\infty} r^{\prime} V\left(r^{\prime}\right) \sin \left(q r^{\prime}\right) d r^{\prime}$$

and find the relation between $q$ and $\theta$.

Calculate this scattering amplitude for the potential $V(r)=V_{0} e^{-r}$ where $V_{0}$ is a constant, and show that at high energies the particles emerge predominantly in a narrow cone around the forward beam direction. Estimate the angular width of the cone.