Exercises

1. Calculate the Clebsch-Gordon coefficients for adding spin one-half to spin one.

2. An electron in a hydrogen atom occupies the combined spin and position state whose spinor-wavefunction is
   $$\chi(r,\theta,\varphi) = R_{2\,1}(r)\,\left[\sqrt{1/3}\,Y_{1\,0}(\theta,\varphi)\,\chi_+ + \sqrt{2/3}\,Y_{1\,1}(\theta,\varphi)\,\chi_-\right].$$
   Here, $\chi_\pm$ are the eigenstates of $S_z$ corresponding to the eigenvalues $\pm \hbar/2$ , respectively, and $r$ , $\theta$ , $\varphi$ are conventional spherical coordinates.
   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 $r$ , $\theta$ , $\varphi$ ?
   8. What is the probability density for finding the electron in the spin-up state (with respect to the $z$ -axis) at radius $r$ ?
   [61]

3. In a low energy neutron-proton system (with zero orbital angular momentum) the potential energy is given by
   $$V({\bf x}) = V_1(r) + V_2(r)\left[3\,\frac{(\mbox{\boldmath $\sig... ...\right] + V_3(r)\,\mbox{\boldmath $\sigma$}_n\cdot\mbox{\boldmath $\sigma$}_p,$$
   where ${\bf x}$ is the vector connecting the two particles, $r=\vert{\bf x}\vert$ , $\sigma$ $_n$ denotes the vector of the Pauli matrices of the neutron, and $\sigma$ $_p$ denotes the vector of the Pauli matrices of the proton. Calculate the potential energy for the neutron-proton system:
   1. In the spin singlet (i.e., spin zero) state.
   2. In the spin triplet (i.e., spin one) state.
   [Hint: Calculate the expectation value of $V({\bf x})$ with respect to the overall spin state.] [53]

4. Consider two electrons in a spin singlet (i.e., spin zero) 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 $S_x=-\hbar/2$ ?
   3. Finally, if electron 1 is in a spin state described by $\cos\alpha_1\,\chi_{z+} + \sin\alpha_1\,{\rm e}^{\,{\rm i}\,\beta_1}\,\chi_{z-}$ , and electron 2 is in a spin state described by $\cos\alpha_2\,\chi_{z+} + \sin\alpha_2\,{\rm e}^{\,{\rm i}\,\beta_2}\,\chi_{z-}$ , what is the probability that the two-electron spin state is a triplet (i.e., spin one) state? Here, $\chi_{z\pm}$ are the eigenstates of $S_z$ corresponding to the eigenvalues, $\pm \hbar/2$ , respectively, for the electron in question. [53]