10.E: Addition of Angular Momentum (Exercises)

1. 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 $r$?
2. 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.
3. 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)

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