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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 ParseError: invalid DekiScript (click for details) Callstack: at (Bookshelves/Quantum_Mechanics/Book:_Introductory_Quantum_Mechanics_(Fitzpatrick)/14:_Scattering_Theory/14.E:_Scattering_Theory_(Exercises)), /content/body/ul/li[5]/p[1]/span, line 1, column 1  {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.
