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# 7.E: Orbital Angular Momentum (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 $$H$$?
