Exercises

1. An electron in a hydrogen atom occupies the combined spin and position state
   
   $$R_{2,1} \left(\sqrt{1/3} Y_{1,0} \chi_+ + \sqrt{2/3} Y_{1,1} \chi_-\right).$$
   
   
   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
   
   $$V(r) = V_1(r) + V_2(r)\left(3 \frac{(\mbox{\boldmath $\sigm... ... \mbox{\boldmath $\sigma$}_1\cdot\mbox{\boldmath $\sigma$}_2,$$
   
   
   where $\mbox{\boldmath$\sigma$}_1$ denotes the vector of the Pauli matrices of the neutron, and $\mbox{\boldmath$\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?