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  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 $H$?

Richard Fitzpatrick 2010-07-20