7.E: Orbital Angular Momentum (Exercises)

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A system is in the state $\psi=Y_{l,m}(\theta,\phi)$. Calculate $\langle L_x\rangle$ and $\langle L_x^{\,2}\rangle$.
Find the eigenvalues and eigenfunctions (in terms of the angles $\theta$ and $\phi$) of $L_x$.
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 $-\hbar$.
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 $H$?

Contributors and Attributions

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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