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8.P: Exercises

  • Page ID
    1237
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    1. Demonstrate that \( {\bf p}\cdot{\bf A}={\bf A}\cdot{\bf p}\) when \( \nabla\cdot{\bf A} = 0\) , where \( {\bf p}\) is the momentum operator, and \( {\bf A}({\bf x})\) is a real function of the position operator, \( {\bf x}\) . Hence, show that the Hamiltonian \ref{870} is Hermitian.
    2. Find the selection rules for the matrix elements \( \langle n,l,m\vert\,x\,\vert n',l',m'\rangle\) , \( \langle n,l,m\vert\,y\,\vert n',l',m'\rangle\) , and \( \langle n,l, m\vert\,z\,\vert n',l',m'\rangle\) to be non-zero. Here, \( \vert n,l,m\rangle\) denotes an energy eigenket of a hydrogen-like atom corresponding to the conventional quantum numbers, \( n\) , \( l\) , and \( m\) .
    3. Demonstrate that \( \left\langle \vert\mbox{\boldmath\) where the average is taken over all directions of the incident radiation.
    4. Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any 2p state to a 1s state of a hydrogen atom is $ w_{2p\rightarrow 1s} = \left(\frac{2}{3}\right)^8\alpha^5\,\frac{m_e\,c^2}{\hbar}=6.26\times 10^8\,{\rm s}^{-1},
$ where \( \alpha\) is the fine structure constant. Hence, deduce that the natural line width of the associated spectral line is $ \frac{{\mit\Delta}\lambda}{\lambda} \simeq 4\times 10^{-8}.
$ The only non-zero \( 1s\leftrightarrow 2p\) electric dipole matrix elements take the values
    5. \( \langle 1,0,0\vert\,x\,\vert 2,1,\pm 1\rangle\) \( = \pm\frac{2^7}{3^5}\,a_0,\) \( \langle 1,0,0\vert\,y\,\vert 2,1,\pm 1\rangle\) \( = {\rm i}\,\frac{2^7}{3^5}\,a_0,\) \( \langle 1,0,0\vert\,z\,\vert 2,1,0\rangle\) \( = \sqrt{2}\,\frac{2^7}{3^5}\,a_0,\) where \( a_0\) is the Bohr radius.

    Contributors

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 8.P: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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