9.P: Exercises

1. Consider a scattering potential of the form Calculate the differential scattering cross-section, $V_0$ for $r<R$ , and is zero for $r>R$ , where $\vert V_0\vert\ll E=\hbar^2\,k^2/2\,m$ , and $k\,R\ll 1$ , the differential cross-section is isotropic, and that the total cross-section is Suppose that the energy is slightly raised. Show that the angular distribution can then be written in the form Obtain an approximate expression for $\delta$ -shell potential: where $s$ -wave phase-shift, $k$ (where $\gamma\gg R^{-1}$ , $\tan(k\,R)$ is not close to zero then the $\tan(k\,R)$ is close to zero then resonance behavior is possible: i.e., $k$ increases. Determine the approximate positions of the resonances (retaining terms up to order $R$ . Obtain an approximate expression for the resonance width Show that the resonances become extremely sharp as $\gamma\rightarrow \infty$ .
2. Show that the differential cross-section for the elastic scattering of a fast electron by the ground-state of a hydrogen atom is where $a_0$ is the Bohr radius.

Contributors

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

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