7.E: Orbital Angular Momentum (Exercises)
 Page ID
 15919

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