9.P: Exercises
- Page ID
- 1247
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- 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 \( \gamma\rightarrow \infty\) .
- 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)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)