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9.P: Exercises

  1. Consider a scattering potential of the form

     

    $\displaystyle 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

     

    $\displaystyle \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

     

    $\displaystyle \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:

     

    $\displaystyle 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

     

    $\displaystyle {\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

     

    $\displaystyle \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.

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