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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 \[H = \frac{L_x^{\,2}+L_y^{\,2}}{2\,I_1} + \frac{L_z^{\,2}}{2\,I_2}.\] What are the eigenvalues of \(H\)?

    Contributors and Attributions

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)