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# 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?

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