Exercises

1. Justify Equation (10.5).

2. Justify Equation (10.7).

3. Consider the Green's function for the Helmholtz equation:
   $$(\nabla^{\,2} + k^{\,2})\,G({\bf x}, {\bf x}') = \delta^{\,3}({\bf x} -{\bf x}').$$
   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
      $$G(R)= -\frac{{\rm e}^{\pm{\rm i}\,k\,R}}{4\pi\,R}.$$

   2. Show that
      $$\int_V dV\, (\nabla^{\,2} +k^{\,2})\,G(R)=1,$$
      where $V$ is a sphere of vanishing radius centered on $R=0$ . Hence, deduce that $G(R)$ is the required Green's function.

4. 1. Demonstrate that
      $$\frac{1}{2\pi}\int_{-\infty}^{\infty} dx\,\exp\left(-\frac{x^2}{2... ...\pi\,\sigma_k^{\,2})^{1/2}}\,\exp \left(-\frac{k^2}{2\,\sigma_k^{\,2}}\right),$$
      where $\sigma_k=1/\sigma_x$ .

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

   3. Finally, by taking the limit $\sigma_x\rightarrow \infty$ , conclude that
      $$\frac{1}{2\pi}\int_{-\infty}^{\infty} dx\,\cos(k\,x)=\delta(x),$$
      where $\delta(x)$ is a Dirac delta function.

5. Demonstrate that the probability current associated with the spherical waves
   $$\psi^\pm({\bf x}) = \frac{\exp(\pm {\rm i}\,k\,r)}{r},$$
   where $r=\vert{\bf x}\vert$ , is
   $${\bf j} = \pm \frac{\hbar\,k}{m}\,\frac{\nabla r}{r^{\,2}}.$$
   Hence, deduce that $\psi^+({\bf x})$ and $\psi^-({\bf x})$ correspond to outgoing and ingoing spherical waves, respectively.

6. Justify Equation (10.37).

7. Show that, in the Born approximation, the total scattering cross-section associated with the Yukawa potential, (10.35), is
   $$\sigma_{\rm total}=\left(\frac{2\,m \,V_0}{ \hbar^{\,2}}\right)^2 \frac{4\pi}{\mu^{\,4}\,(4\,k^{\,2}+\mu^{\,2})} .$$

8. 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
   $$\sigma_{\rm total}= \left(\frac{\sqrt{\pi}\,m\,V_0\,a^{\,3}}{2\,\... ...}\right)^2 2\pi\left[\frac{1-{\rm e}^{-2\,(k\,a)^{\,2}}}{(k\,a)^{\,2}}\right].$$

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

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

11. Demonstrate that, for a spherically symmetric scattering potential,
    $$\sigma_{\rm total} \simeq \frac{m^{\,2}}{\pi\,\hbar^{\,4}}\int d^... ...\bf x}-{\bf x}'\vert]}{k^{\,2}\,\vert{\bf x}-{\bf x}'\vert^{\,2}}\,V(r)\,V(r')$$
    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 $\sigma_{\rm total}$ .]

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

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

14. 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 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

    $$k\,a = n\,\pi-\frac{k}{\gamma} + \frac{k^{\,2}}{\gamma^{\,2}}\,y,$$
    where $y\sim{\cal O}(1)$ . Demonstrate that the S-wave contribution to the total scattering cross-section takes the form
    $$\sigma_0 \simeq \frac{4\pi}{k_n^{\,2}}\,\frac{1}{1+y^{\,2}} = \fr... ...{\,2}}\,\frac{{\mit\Gamma}_n^{\,2}/4}{(E-E_n)^{\,2} + {\mit\Gamma}_n^{\,2}/4}.$$
    where

    $$k_n \simeq \frac{n\,\pi}{a},$$

    $$E_n \simeq \frac{n^{\,2}\,\pi^{\,2}\,\hbar^{\,2}}{2\,m\,a^{\,2}},$$

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

    
    

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

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

    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.

15. 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
    $$\frac{d\sigma}{d{\mit\Omega}}\simeq \left(\frac{m_p\,e^{\,2}}{16\... ...frac{1}{\cos^4(\theta/2)}-\frac{1}{\sin^2(\theta/2)\,\cos^2(\theta/2)}\right],$$
    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.