8.P: Exercises
- Page ID
- 1237
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- 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.
- 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 \( m\) .
- Demonstrate that \( \left\langle \vert\mbox{\boldmath\) where the average is taken over all directions of the incident radiation.
- Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any 2p state to a 1s state of a hydrogen atom is where \( \alpha\) is the fine structure constant. Hence, deduce that the natural line width of the associated spectral line is The only non-zero \( 1s\leftrightarrow 2p\) electric dipole matrix elements take the values
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)