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14.E: Scattering Theory (Exercises)

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    • Show that, in the Born approximation, the total scattering cross-section 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 cross-section for the elastic scattering of a fast electron by the ground state of a hydrogen atom is \[\frac{d\sigma}{d{\mit\Omega}} = \left(\frac{2\,m_e\,e^{\,2}}{4\pi\,\epsilon_0\,\hbar^{\,2}\,q^{\,2}}\right)^2\left(1-\frac{16}{[4+(q\,a_0)^{\,2}]^{\,2}}\right)^2,\] 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: \[V(r) = \left(\frac{\hbar^{\,2}}{2\,m}\right)\gamma\,\delta(r-a),\] where \(\gamma, a >0\). Show that the \(S\)-wave phase-shift is given by \[\delta_0 = -k\,a + \tan^{-1}\left[\frac{1}{\cot(k\,a)+\gamma/k}\right].\] 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 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

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

      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.

    Contributors and Attributions

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)