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Next: Relativistic Electron Theory Up: Scattering Theory Previous: Scattering of Identical Particles


  1. Justify Equation (10.5).

  2. Justify Equation (10.7).

  3. Consider the Green's function for the Helmholtz equation:

    $\displaystyle (\nabla^{\,2} + k^{\,2})\,G({\bf x}, {\bf x}') = \delta^{\,3}({\bf x} -{\bf x}').$

    1. Assuming, as seems reasonable, that $ G({\bf x},{\bf x}') = G(R)$ , where $ {\bf R} = {\bf x}-{\bf x}'$ , demonstrate that the two independent solutions in the region $ R>0$ are

      $\displaystyle G(R)= -\frac{{\rm e}^{\pm{\rm i}\,k\,R}}{4\pi\,R}.

    2. Show that

      $\displaystyle \int_V dV\, (\nabla^{\,2} +k^{\,2})\,G(R)=1,

      where $ V$ is a sphere of vanishing radius centered on $ R=0$ . Hence, deduce that $ G(R)$ is the required Green's function.

    1. Demonstrate that

      $\displaystyle \frac{1}{2\pi}\int_{-\infty}^{\infty} dx\,\exp\left(-\frac{x^2}{2...
...\pi\,\sigma_k^{\,2})^{1/2}}\,\exp \left(-\frac{k^2}{2\,\sigma_k^{\,2}}\right),

      where $ \sigma_k=1/\sigma_x$ .

    2. Show that

      $\displaystyle \frac{1}{(2\pi\,\sigma_k^{\,2})^{1/2}}\int_{-\infty}^{\infty} dk\,\exp \left(-\frac{k^2}{2\,\sigma_k^{\,2}}\right) = 1.

    3. Finally, by taking the limit $ \sigma_x\rightarrow \infty$ , conclude that

      $\displaystyle \frac{1}{2\pi}\int_{-\infty}^{\infty} dx\,\cos(k\,x)=\delta(x),

      where $ \delta(x)$ is a Dirac delta function.

  4. Demonstrate that the probability current associated with the spherical waves

    $\displaystyle \psi^\pm({\bf x}) = \frac{\exp(\pm {\rm i}\,k\,r)}{r},

    where $ r=\vert{\bf x}\vert$ , is

    $\displaystyle {\bf j} = \pm \frac{\hbar\,k}{m}\,\frac{\nabla r}{r^{\,2}}.

    Hence, deduce that $ \psi^+({\bf x})$ and $ \psi^-({\bf x})$ correspond to outgoing and ingoing spherical waves, respectively.

  5. Justify Equation (10.37).

  6. Show that, in the Born approximation, the total scattering cross-section associated with the Yukawa potential, (10.35), is

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

  7. Consider a scattering potential of the form

    $\displaystyle V(r)=V_0\,\exp\left(-\frac{r^{\,2}}{a^{\,2}}\right).

    Demonstrate, using the Born approximation, that

    $\displaystyle \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],


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

  8. 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...

    where $ q=\vert{\bf k}-{\bf k}'\vert$ , and $ a_0$ is the Bohr radius.

  9. Justify Equations (10.57) and (10.59).

  10. Demonstrate that, for a spherically symmetric scattering potential,

    $\displaystyle \sigma_{\rm total} \simeq \frac{m^{\,2}}{\pi\,\hbar^{\,4}}\int d^...
...\bf x}-{\bf x}'\vert]}{k^{\,2}\,\vert{\bf x}-{\bf x}'\vert^{\,2}}\,V(r)\,V(r')

    in each of the following ways:
    1. By directly integrating the differential scattering cross-section obtained from the Born approximation.
    2. By utilizing the optical theorem in combination with the first two terms in the Born expansion. [Note that the first term in this expansion is real, and, therefore, does not contribute to $ \sigma_{\rm total}$ .]

  11. Justify Equations (10.98) and (10.100).

  12. 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\equiv \hbar^{\,2}\,k^{\,2}/2\,m$ , and $ k\,R\ll 1$ ,

    $\displaystyle \frac{d\sigma}{d{\mit\Omega}}=\left(\frac{4}{9}\right)\left(\frac...
...ght)\left[1+\frac{2}{5}\,(k\,R)^{\,2}\,\cos\theta+{\cal O}(k\,R)^{\,4}\right],


    $\displaystyle \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].

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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle \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 10.8.) 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

    $\displaystyle 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

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


    $\displaystyle k_n$ $\displaystyle \simeq \frac{n\,\pi}{a},$    
    $\displaystyle E_n$ $\displaystyle \simeq \frac{n^{\,2}\,\pi^{\,2}\,\hbar^{\,2}}{2\,m\,a^{\,2}},$    
    $\displaystyle {\mit\Gamma}_n$ $\displaystyle \simeq \frac{4\,n\,\pi\,E_n}{(\gamma\,a)^{\,2}}.$    

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

    $\displaystyle \sigma_0\simeq \frac{4\pi}{k^{\,2}}\left(\sin^2(k\,a)+\sum_{n=1,\...
...}\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.

  14. Consider the mutual scattering of two, counter-propagating, unpolarized, proton beams in the center of mass frame. Making use of the Born approximation, demonstrate that the differential scattering cross-section is

    $\displaystyle \frac{d\sigma}{d{\mit\Omega}}\simeq \left(\frac{m_p\,e^{\,2}}{16\...

    where $ \theta$ is the scattering angle, $ m_p$ the proton mass, and $ \hbar\, k$ the magnitude of the incident momenta of protons in each beam.

next up previous
Next: Relativistic Electron Theory Up: Scattering Theory Previous: Scattering of Identical Particles
Richard Fitzpatrick 2016-01-22