Exercises

1. Consider a scattering potential of the form
   $$V(r)=V_0\,\exp(-r^2/a^2).$$
   Calculate the differential scattering cross-section, $d\sigma/d{\mit\Omega}$ , using the Born approximation.

2. Consider a scattering potential that takes the constant value $V_0$ for $r<R$ , and is zero for $r>R$ , where $V_0$ may be either positive or negative. Using the method of partial waves, show that for $\vert V_0\vert\ll E=\hbar^2\,k^2/2\,m$ , and $k\,R\ll 1$ , the differential cross-section is isotropic, and that the total cross-section is
   $$\sigma_{\rm tot} = \left(\frac{16\pi}{9}\right) \frac{m^2\,V_0^{\,2}\,R^{\,6}}{\hbar^4}.$$
   Suppose that the energy is slightly raised. Show that the angular distribution can then be written in the form
   $$\frac{d\sigma}{d{\mit\Omega} }= A + B\,\cos\theta.$$
   Obtain an approximate expression for $B/A$ .

3. Consider scattering by a repulsive $\delta$ -shell potential:
   $$V(r) = \left(\frac{\hbar^2}{2\,m}\right)\gamma\,\delta(r-R),$$
   where $\gamma>0$ . Find the equation that determines the $s$ -wave phase-shift, $\delta_0$ , as a function of $k$ (where $E=\hbar^2\,k^2/2\,m$ ). Assume that $\gamma\gg R^{-1}$ , $k$ . Show that if $\tan(k\,R)$ is not close to zero then the $s$ -wave phase-shift resembles the hard sphere result discussed in the text. Furthermore, show that if $\tan(k\,R)$ is close to zero then resonance behavior is possible: i.e., $\cot \delta_0$ goes through zero from the positive side as $k$ increases. Determine the approximate positions of the resonances (retaining terms up to order $1/\gamma$ ). Compare the resonant energies with the bound state energies for a particle confined within an infinite spherical well of radius $R$ . Obtain an approximate expression for the resonance width
   $${\mit\Gamma} = - \frac{2}{[d(\cot\delta_0)/dE]_{E=E_r}}.$$
   Show that the resonances become extremely sharp as $\gamma\rightarrow \infty$ .

4. Show that the differential cross-section for the elastic scattering of a fast electron by the ground-state of a hydrogen atom is
   $$\frac{d\sigma}{d{\mit\Omega}} = \left(\frac{2\,m_e\,e^2}{4\pi\,\e... ...n_0\,\hbar^{\,2}\,q^2}\right)^2\left(1-\frac{16}{[4+(q\,a_0)^2]^{\,2}}\right),$$
   where $q=\vert{\bf k}-{\bf k}'\vert$ , and $a_0$ is the Bohr radius.