6.P: Exercises
- Page ID
- 1215
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- Calculate the Clebsch-Gordon coefficients for adding spin one-half to spin one.
- Calculate the Clebsch-Gordon coefficients for adding spin one to spin one.
- An electron in a hydrogen atom occupies the combined spin and position state whose wavefunction is \[\psi = R_{2\,1}(r)\,\left[\sqrt{1/3}\,Y_{1\,0}(\theta,\varphi)\,\chi_+ + \sqrt{2/3}\,Y_{1\,1}(\theta,\varphi)\,\chi_-\right].\]
- What values would a measurement of \( L^2\) yield, and with what probabilities?
- Same for \( S^2\) .
- Same for \( J^{\,2}\) .
- Same for \( r\) , \( \varphi\) ?
- What is the probability density for finding the electron in the spin up state (with respect to the \( r\) ?
- In a low energy neutron-proton system (with zero orbital angular momentum) the potential energy is given by where \( \sigma\) \( \sigma\) \( V({\bf x})\) with respect to the overall spin state.]
- Consider two electrons in a spin singlet (i.e., spin zero) state.
- If a measurement of the spin of one of the electrons shows that it is in the state with \( z\) -component of the spin of the other electron yields \( S_y=\hbar/2\) , what is the probability that a measurement of the \( S_x=-\hbar/2\) ?
- Finally, if electron 1 is in a spin state described by , and electron 2 is in a spin state described by , what is the probability that the two-electron spin state is a triplet (i.e., spin one) state?
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)