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7.E: Orbital Angular Momentum (Exercises)

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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

      \[H = \frac{L_x^2+L_y^2}{2\,I_1} + \frac{L_z^2}{2\,I_2}. \nonumber \]

      What are the eigenvalues of \(H\)?

    This page titled 7.E: Orbital Angular Momentum (Exercises) was last modified on Fri, 13 Mar 2026 03:53:12 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.