Problems

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 $r$?
   [from Griffiths]

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.
   [from Gaziorowicz]

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? [from Gaziorowicz]