# 10.E: Addition of Angular Momentum (Exercises)

- Page ID
- 15788

- 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)

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