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\centerline{\bf Quantum Calculations on a Ring I}
\bigskip

%Before you begin, recall that an arbitrary state $\left|\Phi\right\rangle$ can be written as
%
%$$ \left| \Phi\right\rangle
%\doteq \pmatrix{\vdots\cr \langle 4|\Phi\rangle \cr \langle 3|\Phi\rangle \cr \langle 2|\Phi\rangle \cr \langle 1|\Phi\rangle \cr \langle 0|\Phi\rangle \cr \langle -1|\Phi\rangle \cr \langle -2|\Phi\rangle \cr \langle -3|\Phi\rangle \cr \langle -4|\Phi\rangle \cr\vdots} = \pmatrix{\vdots\cr a_{4} \cr a_{3} \cr a_{2} \cr a_{1} \cr a_{0} \cr a_{-1} \cr a_{-2} \cr a_{-3} \cr a_{-4}\cr\vdots}  $$

In this activity, your group will carry out calculations on each of the following normalized abstract quantum states on a ring:

$$\left| \Phi_a\right\rangle
= \sqrt{ 2\over 12}\left| 3\right\rangle
+ \sqrt{ 1\over 12}\left| 2\right\rangle
+\sqrt{ 3\over 12}\left| 0\right\rangle
+\sqrt{ 2\over 12}\left| -1\right\rangle
+\sqrt{ 1\over 12}\left| -3\right\rangle
+\sqrt{ 3\over 12}\left| -4\right\rangle $$

$$ \left| \Phi_b\right\rangle
\doteq \pmatrix{0 \cr \sqrt{ 2\over 12} \cr \sqrt{ 1\over 12} \cr 0 \cr \sqrt{ 3\over 12} \cr \sqrt{ 2\over 12} \cr 0 \cr \sqrt{ 1\over 12} \cr \sqrt{ 3\over 12}}$$

%$$ \Phi_c(\phi)
%= \sqrt {1\over {24 \pi}} \left[ \sqrt{2} \left(e^{i 3 \phi} +e^{-i 1 \phi}\right)
%+ \left( e^{i 2 \phi} + e^{-i 3 \phi}\right)
%+ \sqrt{3}\left(1 + e^{-i 4 \phi}\right) \right] $$

$$ \Phi_c(\phi)
\doteq \sqrt {1\over {24 \pi r_0}} \left(\sqrt{2} e^{i 3 \phi} + e^{i 2 \phi}+ \sqrt{3}+\sqrt{2} e^{-i 1 \phi}
 + e^{-i 3 \phi} + \sqrt{3} e^{-i 4 \phi} \right) $$

%$$ \Phi_c(\phi)
%= \sqrt {2\over {24 \pi}} e^{i 3 \phi} + \sqrt {1\over {24 \pi}}  e^{i 2 \phi}+\sqrt {3\over {24 \pi}} +\sqrt {2\over {24 \pi}} e^{-i 1 \phi} + \sqrt {1\over {24 \pi}} e^{-i 3 \phi} + \sqrt {3\over {24 \pi}} e^{-i 4 \phi}$$

For each of the following questions, state the postulate of quantum mechanics you use to complete the calculation and show explicitly how you use that postulate to answer the question.

\begin{enumerate}
\item For each state above, what is the probability that you would measure the $z$-component of angular momentum to be $-4\hbar$? $0\hbar$? $-2\hbar$? $3\hbar$?

\vfill

\item  What other possible values for the $z$-component of angular momentum could you have obtained with non-zero probability?
\vfill

\newpage
\item For each state, what is the probability that you would measure the energy to be $16\hbar^2 \over {2I}$? 0? $4\hbar^2 \over {2I}$? $9\hbar^2 \over {2I}$?

\vfill

\item  If you measured the energy, what possible values could you have obtained with non-zero probability?
\vfill

%\item  If you measured the for each state, what is the
%probability that you would obtain:
%\tabskip=2pc
%\halign{\hfil#\hfil &\hfil#\hfil &\hfil#\hfil &\hfil#\hfil \cr
%Group 1: $2\hbar$ & Group 2: $-4\hbar$ & Group 3: $-3\hbar$ & Group 4: $4\hbar$\cr
%Group 5: $-1\hbar$ & Group 6: $0$ & Group 7: $-2\hbar$ & Group 8: $3\hbar$ \cr}
%
%\item  If you measured the energy for each state, what is the
%probability that you would obtain:
%\tabskip=2pc
%\halign{\hfil#\hfil &\hfil#\hfil &\hfil#\hfil &\hfil#\hfil \cr
%Group 1: ${9 \hbar^2}\over{2 I}$ & Group 2: $0$ & Group 3: ${4 \hbar^2}\over{2 I}$ & Group 4: ${1 \hbar^2}\over{2 I}$\cr
%Group 5: ${16 \hbar^2}\over{2 I}$ & Group 6: ${1 \hbar^2}\over{2 I}$ & Group 7: ${9 \hbar^2}\over{2 I}$ & Group 8: ${4 \hbar^2}\over{2 I}$ \cr}

\item How are the calculations you made for the different state representations similar and different from each other? Be prepared to compare and contrast the calculations you made for each of the different representations (ket, matrix, wavefunction).
\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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