Exercises

1. A system is in the state $\psi=Y_{l,m}(\theta,\phi)$. Calculate $\langle L_x\rangle$ and $\langle L_x^{ 2}\rangle$.

2. Find the eigenvalues and eigenfunctions (in terms of the angles $\theta$ and $\phi$) of $L_x$.

3. Consider a beam of particles with $l=1$. A measurement of $L_x$ yields the result $\hbar$. What values will be obtained by a subsequent measurement of $L_z$, and with what probabilities? Repeat the calculation for the cases in which the measurement of $L_x$ yields the results $0$ and $-\hbar$.

4. The Hamiltonian for an axially symmetric rotator is given by
   
   $$H = \frac{L_x^{ 2}+L_y^{ 2}}{2 I_1} + \frac{L_z^{ 2}}{2 I_2}.$$
   
   
   What are the eigenvalues of $H$?