\documentclass[10pt]{article}
\usepackage{graphicx, multicol,wrapfig,exscale,epsfig,fancybox,fullpage}
\pagestyle{empty}

\parindent=0pt
\parskip=.1in

\newcommand\hs{\hspace{6pt}}

\begin{document}

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

$$\vert \Phi\rangle=\sqrt{2\over 3}\vert -3\rangle+ {1 \over \sqrt{6}}\vert -1\rangle+{i \over \sqrt{6}}\vert 3\rangle$$

\begin{enumerate}
\item What is the expectation value of $\hat{L}_z$ as a function of time for this state?
\vfill

\item What is the expectation value of the energy as a function of time for this state?
\vfill
\end{enumerate}

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