14.E: Scattering Theory (Exercises)
 Page ID
 15980
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Show that, in the Born approximation, the total scattering crosssection associated with the Yukawa potential, ([e10.35ee]), is \[\sigma_{\rm total}=\left(\frac{2\,m \,V_0}{ \hbar^{\,2}}\right)^2 \frac{4\pi}{\mu^{\,4}\,(4\,k^{\,2}+\mu^{\,2})} .\]

Consider a scattering potential of the form \[V(r)=V_0\,\exp\left(\frac{r^{\,2}}{a^{\,2}}\right).\] Demonstrate, using the Born approximation, that \[\frac{ d\sigma}{d{\mit\Omega}}=\left(\frac{\sqrt{\pi}\,m\,V_0\,a^{\,3}}{2\,\hbar^{\,2}}\right)^2\exp\left[2\,(k\,a)^{\,2}\,\sin^2(\theta/2)\right],\] and \[\sigma_{\rm total}= \left(\frac{\sqrt{\pi}\,m\,V_0\,a^{\,3}}{2\,\hbar^{\,2}}\right)^2 2\pi\left[\frac{1{\rm e}^{2\,(k\,a)^{\,2}}}{(k\,a)^{\,2}}\right].\]

Show that the differential crosssection for the elastic scattering of a fast electron by the ground state of a hydrogen atom is \[\frac{d\sigma}{d{\mit\Omega}} = \left(\frac{2\,m_e\,e^{\,2}}{4\pi\,\epsilon_0\,\hbar^{\,2}\,q^{\,2}}\right)^2\left(1\frac{16}{[4+(q\,a_0)^{\,2}]^{\,2}}\right)^2,\] where \(q={\bf k}{\bf k}'\), and \(a_0\) is the Bohr radius.

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 \(k\,R\ll 1\), \[\frac{d\sigma}{d{\mit\Omega}}=\left(\frac{4}{9}\right)\left(\frac{m^{\,2}\,V_0^{\,2}\,R^{\,6}}{\hbar^{\,4}}\right)\left[1+\frac{2}{5}\,(k\,R)^{\,2}\,\cos\theta+{\cal O}(k\,R)^{\,4}\right],\] and \[\sigma_{\rm total} = \left(\frac{16\,\pi}{9}\right)\left(\frac{m^{\,2}\,V_0^{\,2}\,R^{\,6}}{\hbar^{\,4}}\right)\left[1+{\cal O}(k\,R)^{\,4}\right].\]

Consider scattering of particles of mass \(m\) and incident wavenumber \(k\) by a repulsive \(\delta\)shell potential: \[V(r) = \left(\frac{\hbar^{\,2}}{2\,m}\right)\gamma\,\delta(ra),\] where \(\gamma, a >0\). Show that the \(S\)wave phaseshift is given by \[\delta_0 = k\,a + \tan^{1}\left[\frac{1}{\cot(k\,a)+\gamma/k}\right].\] 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 \[\delta_0 \simeq k\,a +\frac{k}{\gamma}  \left(\frac{k}{\gamma}\right)^{\,2}\cot(k\,a) + {\cal O}\left(\frac{k}{\gamma}\right)^{\,3}.\] 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 \[k\,a = n\,\pi\frac{k}{\gamma} + \frac{k^{\,2}}{\gamma^{\,2}}\,y,\] where \(y\sim{\cal O}(1)\). Demonstrate that the \(S\)wave contribution to the total scattering crosssection takes the form \[\sigma_0 \simeq \frac{4\pi}{k_n^{\,2}}\,\frac{1}{1+y^{\,2}} = \frac{4\pi}{k_n^{\,2}}\,\frac{{\mit\Gamma}_n^{\,2}/4}{(EE_n)^{\,2} + {\mit\Gamma}_n^{\,2}/4}.\] where \[\begin{aligned} k_n & \simeq \frac{n\,\pi}{a},\\[0.5ex] E_n &\simeq \frac{n^{\,2}\,\pi^{\,2}\,\hbar^{\,2}}{2\,m\,a^{\,2}},\\[0.5ex] {\mit\Gamma}_n &\simeq \frac{4\,n\,\pi\,E_n}{(\gamma\,a)^{\,2}}.\end{aligned}\]
Hence, deduce that the net \(S\)wave contribution to the total scattering crosssection is \[\sigma_0\simeq \frac{4\pi}{k^{\,2}}\left(\sin^2(k\,a)+\sum_{n=1,\infty}\frac{{\mit\Gamma}_n^{\,2}/4}{(EE_n)^{\,2} + {\mit\Gamma}_n^{\,2}/4}\right).\]
Obviously, there are resonant contributions to the crosssection 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.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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