10.E: Addition of Angular Momentum (Exercises)

Last updated
Save as PDF

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}}$

An electron in a hydrogen atom occupies the combined spin and position state \[R_{2,1}(r)\,\left[\sqrt{1/3}\,Y_{1,0}(\theta,\phi)\,\chi_+ + \sqrt{2/3}\,Y_{1,1}(\theta,\phi)\,\chi_-\right].\]
1. What values would a measurement of $L^2$ yield, and with what probabilities?
2. Same for $L_z$.
3. Same for $S^{\,2}$.
4. Same for $S_z$.
5. Same for $J^{\,2}$.
6. Same for $J_z$.
7. What is the probability density for finding the electron at $r$, $\theta$, $\phi$?
8. 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:
1. In the spin singlet state.
2. In the spin triplet state.
Consider two electrons in a spin singlet state.
1. 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 $S_z=\hbar/2$?
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$}}$