10.E: Addition of Angular Momentum (Exercises)
- Page ID
- 15788
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- An electron in a hydrogen atom occupies the combined spin and position state \[R_{2,1}(r)\,\left[\sqrt{1/3}\,Y_{1,0}(\theta,\phi)\,\chi_+ + \sqrt{2/3}\,Y_{1,1}(\theta,\phi)\,\chi_-\right].\]
- What values would a measurement of \(L^2\) yield, and with what probabilities?
- Same for \(L_z\).
- Same for \(S^{\,2}\).
- Same for \(S_z\).
- Same for \(J^{\,2}\).
- Same for \(J_z\).
- What is the probability density for finding the electron at \(r\), \(\theta\), \(\phi\)?
- What is the probability density for finding the electron in the spin up state (with respect to the \(z\)-axis) at radius \(r\)?
- In a low energy neutron-proton system (with zero orbital angular momentum), the potential energy is given by \[V(r) = V_1(r) + V_2(r)\left[3\,\frac{(\sigma_1\cdot{\bf r})\,(\sigma_2\cdot {\bf r})}{r^2} -\sigma_1\cdot\sigma_2\right] + V_3(r)\,\sigma_1\cdot\sigma_2,\] where \(\sigma_1\) denotes the vector of the Pauli matrices of the neutron, and \(\sigma_2\) denotes the vector of the Pauli matrices of the proton. Calculate the potential energy for the neutron-proton system:
- In the spin singlet state.
- In the spin triplet state.
- Consider two electrons in a spin singlet state.
- If a measurement of the spin of one of the electrons shows that it is in the state with \(S_z=\hbar/2\), what is the probability that a measurement of the \(z\)-component of the spin of the other electron yields \(S_z=\hbar/2\)?
- If a measurement of the spin of one of the electrons shows that it is in the state with \(S_y=\hbar/2\), what is the probability that a measurement of the \(x\)-component of the spin of the other electron yields \(S_x=-\hbar/2\)?
Finally, if electron 1 is in a spin state described by \(\cos\alpha_1\,\chi_+ + \sin\alpha_1\,{\rm e}^{\,{\rm i}\,\beta_1}\,\chi_-\), and electron 2 is in a spin state described by \(\cos\alpha_2\,\chi_+ + \sin\alpha_2\,{\rm e}^{\,{\rm i}\,\beta_2}\,\chi_-\), what is the probability that the two-electron spin state is a triplet state?
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)