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\centerline{\bf Quantum Calculations on a Ring II}
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In this activity, your group will carry out calculations on the following quantum state on a ring:
%$$\ket{\psi}={3 \over 5}\ket{-1}+{4 \over 5}\ket{2}$$
%$$\vert \Psi\rangle={3 \over 5}\vert -1\rangle+{4 \over 5}\vert 2\rangle$$
$$\vert \Phi\rangle=\sqrt{2\over 3}\vert -3\rangle+ {1 \over \sqrt{6}}\vert -1\rangle+{i \over \sqrt{6}}\vert 3\rangle$$
%$$\left| \Psi\right\rangle
%= \sqrt{ 1\over 4} \left(\left| 1\right\rangle 
%+ \sqrt{2}\left| 2\right\rangle 
%+\left| 3\right\rangle \right) $$ 

\begin{enumerate}
\item You carry out a measurement to determine the z-component of the angular momentum of the particle at time, $t$. Calculate the probability that you measure the z-component of the angular momentum to be $3 \hbar$. What representation/basis did you use to do this calculation and why did you use this representation?
\vfill

\item You carry out a measurement to determine the energy of the particle at time, $t$. Calculate the probability that you measure the energy to be ${9 \hbar^2} \over {2 I}$. What representation/basis did you use to do this calculation and why did you use this representation?
\vfill

%\newpage
%\item Write the state as a function of time.

\item Calculate the probability that the particle can be found in the region $0<\phi< {\pi \over 3}$ at some time, $t$. What representation/basis did you use to do this calculation and why did you use this representation?

%\item  You carry out a measurement on the location of the particle at time, t=0. Calculate the probability that the particle can be found in the region $0<\phi< {\pi \over 2}$. What representation/basis did you use to do this calculation and why did you use this representation?
\vfill

\item Under what circumstances do measurement probabilities change with time?
\vfill

\end{enumerate}

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