14.9: Exercises

1. Show that, in the Born approximation, the total scattering cross-section associated with the Yukawa potential, \ref{e10.35ee}, is

   \[\sigma_{\rm total}=\left(\frac{2\,m \,V_0}{ \hbar^2}\right)^2 \frac{4\pi}{\mu^{\,4}\,(4\,k^2+\mu^2)} . \nonumber \]

2. Consider a scattering potential of the form

   \[V(r)=V_0\,\exp\left(-\frac{r^2}{a^2}\right). \nonumber \]

   Demonstrate, using the Born approximation, that

   \[\frac{ d\sigma}{d\Omega}=\left(\frac{\sqrt{\pi}\,m\,V_0\,a^{\,3}}{2\,\hbar^2}\right)^2\exp\left[-2\,(k\,a)^2\,\sin^2(\theta/2)\right], \nonumber \]

   and

   \[\sigma_{\rm total}= \left(\frac{\sqrt{\pi}\,m\,V_0\,a^{\,3}}{2\,\hbar^2}\right)^2 2\pi\left[\frac{1-\rm e^{-2\,(k\,a)^2}}{(k\,a)^2}\right]. \nonumber \]

3. Show that the differential cross-section for the elastic scattering of a fast electron by the ground state of a hydrogen atom is

   \[\frac{d\sigma}{d\Omega} = \left(\frac{2\,m_e\,e^2}{4\pi\,\epsilon_0\,\hbar^2\,q^2}\right)^2\left(1-\frac{16}{[4+(q\,a_0)^2]^2}\right)^2, \nonumber \]

   where \(q=|{\bf k}-{\bf k}'|\), and \(a_0\) is the Bohr radius.

4. 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 \(|V_0|\ll E\equiv \hbar^2\,k^2/2\,m\), and \(k\,R\ll 1\),

   \[\frac{d\sigma}{d\Omega}=\left(\frac{4}{9}\right)\left(\frac{m^2\,V_0^2\,R^{\,6}}{\hbar^{\,4}}\right)\left[1+\frac{2}{5}\,(k\,R)^2\,\cos\theta+{\cal O}(k\,R)^{\,4}\right], \nonumber \]

   and

   \[\sigma_{\rm total} = \left(\frac{16\,\pi}{9}\right)\left(\frac{m^2\,V_0^2\,R^{\,6}}{\hbar^{\,4}}\right)\left[1+{\cal O}(k\,R)^{\,4}\right]. \nonumber \]

5. Consider scattering of particles of mass \(m\) and incident wavenumber \(k\) by a repulsive \(\delta\)-shell potential:

   \[V(r) = \left(\frac{\hbar^2}{2\,m}\right)\gamma\,\delta(r-a), \nonumber \]

   where \(\gamma, a >0\). Show that the \(S\)-wave phase-shift is given by

   \[\delta_0 = -k\,a + \tan^{-1}\left[\frac{1}{\cot(k\,a)+\gamma/k}\right]. \nonumber \]

   Assuming that \(\gamma\gg k, a^{\,-1}\), demonstrate that if \(\cot(k\,a) \sim{\cal O}(1)\) then the solution of the previous equation takes the form

   \[\delta_0 \simeq -k\,a +\frac{k}{\gamma} - \left(\frac{k}{\gamma}\right)^2\cot(k\,a) + {\cal O}\left(\frac{k}{\gamma}\right)^{\,3}. \nonumber \]

   Of course, in the limit \(\gamma\rightarrow\infty\), the preceding equation yields \(\delta_0=-k\,a\), which is the same result obtained when particles are scattered by a hard sphere of radius \(a\). (See Section 1.7.) This is not surprising, because a strong repulsive \(\delta\)-shell potential is indistinguishable from hard sphere as far as external particles are concerned.

   The previous solution breaks down when \(k\,a\simeq n\,\pi\), where \(n\) is a positive integer. Suppose that

   \[k\,a = n\,\pi-\frac{k}{\gamma} + \frac{k^2}{\gamma^2}\,y, \nonumber \]

   where \(y\sim{\cal O}(1)\). Demonstrate that the \(S\)-wave contribution to the total scattering cross-section takes the form

   \[\sigma_0 \simeq \frac{4\pi}{k_n^2}\,\frac{1}{1+y^2} = \frac{4\pi}{k_n^2}\,\frac{{\mit\Gamma}_n^2/4}{(E-E_n)^2 + {\mit\Gamma}_n^2/4}. \nonumber \]

   where

   \[\begin{align} k_n & \simeq \frac{n\,\pi}{a},\\[4pt] E_n &\simeq \frac{n^2\,\pi^2\,\hbar^2}{2\,m\,a^2},\\[4pt] {\mit\Gamma}_n &\simeq \frac{4\,n\,\pi\,E_n}{(\gamma\,a)^2}.\end{align} \nonumber \]

   Hence, deduce that the net \(S\)-wave contribution to the total scattering cross-section is

   \[\sigma_0\simeq \frac{4\pi}{k^2}\left(\sin^2(k\,a)+\sum_{n=1,\infty}\frac{{\mit\Gamma}_n^2/4}{(E-E_n)^2 + {\mit\Gamma}_n^2/4}\right). \nonumber \]

   Obviously, there are resonant contributions to the cross-section whenever \(E\simeq E_n\). Note that the \(E_n\) are the possible energies of particles trapped within the \(\delta\)-shell potential. Hence, the resonances are clearly associated with incident particles tunneling though the \(\delta\)-shell and forming transient trapped states. However, the width of the resonances (in energy) decreases strongly as the strength, \(\gamma\), of the shell increases.