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\centerline{\bf Quantum Calculations on a Ring III}
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Consider the following normalized abstract quantum state on a ring:

$$\Phi(\phi)={\sqrt{8\over {5\pi r_0}}} \cos^3(2\phi)$$

\begin{enumerate}
\item  If you measured the $z$-component of angular momentum, what is the
probability that you would measure $2\hbar$? $-3\hbar$?
\vfill

\item  If you measured the $z$-component of angular momentum, what other possible values could you have obtained with non-zero probability?
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\item  If you measured the energy, what possible values could you have obtained with non-zero probability?
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\item  What is the probability that the particle can be found in the region $0<\phi< {\pi \over 2}$?
\end{enumerate}

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\leftline{\it by Corinne Manogue, Kerry Browne, Liz Gire, Mary Bridget Kustusch, David McIntyre}
\leftline{\copyright 2012 Corinne A. Manogue}
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