\documentclass[10pt]{article}
\usepackage{graphicx}
\pagestyle{empty}

\parindent=0pt
\parskip=.1in

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

\begin{document}

\centerline{\textbf{Finding the Unknown States Leaving an Oven (Spin-1/2)}}
\bigskip

\begin{enumerate}

\item Start the SPINS program and choose \underline{Unknown \# 1} under the \underline{initialize} menu.  This causes the atoms to leave the oven in a definite quantum state, which we call $\vert \psi_{1} \rangle$.  Now measure the six probabilities $\left| \langle \phi \vert \psi_{1} \rangle \right| ^{2}$, where $\vert \phi \rangle$ corresponds to spin up and spin down along the three axes.  Fill in the table for $\vert \psi_{1} \rangle$ on the worksheet.  Assume that we want to write the unknown state vectors in terms of the $\vert \pm \rangle$ basis, $i.e. \; \vert \psi_{1} \rangle = a\vert + \rangle + b \vert = \rangle$, where $a$ and $b$ are complex coefficients.  We thus must use the data to find the values of $a$ and $b$.

\item Repeat this exercise for \underline{Unknown \# 2} ($\vert \psi_{2} \rangle$ ), \underline{Unknown \# 3} ($\vert \psi_{3} \rangle$ ), and \underline{Unknown \# 4} ($\vert \psi_{4} \rangle$).

\item Design an experiment to verify your results (Hint: recall the general spin 1/2 state vector can be written as $\vert + \rangle{n}=\cos{\frac{\theta}{2}}\vert + \rangle + \sin{\frac{\theta}{2}}e^{i \phi}\vert - \rangle$ ).

\end{enumerate}
\bigskip\bigskip


\setlength{\tabcolsep}{35pt}
\renewcommand{\arraystretch}{2}

\centerline{Unknown $\vert \psi_{1} \rangle$}
\begin{tabular}{|c|c|c|c|} \hline

Probabilities & \multicolumn{3}{c|}{Axis} \\
\hline
Result & x & y & z \\
\hline
Spin up $\uparrow$ &  &  &  \\
\hline
Spin down $\downarrow $ &  &  &  \\
\hline
\end{tabular}

\vfill
\newpage

\centerline{Unknown $\vert \psi_{2} \rangle$}
\begin{tabular}{|c|c|c|c|} \hline

Probabilities & \multicolumn{3}{c|}{Axis} \\
\hline
Result & x & y & z \\
\hline
Spin up $\uparrow$ &  &  &  \\
\hline
Spin down $\downarrow $ &  &  &  \\
\hline
\end{tabular}
\vfill

\centerline{Unknown $\vert \psi_{3} \rangle$}
\begin{tabular}{|c|c|c|c|} \hline

Probabilities & \multicolumn{3}{c|}{Axis} \\
\hline
Result & x & y & z \\
\hline
Spin up $\uparrow$ &  &  &  \\
\hline
Spin down $\downarrow $ &  &  &  \\
\hline
\end{tabular}

\vfill

\centerline{Unknown $\vert \psi_{4} \rangle$}
\begin{tabular}{|c|c|c|c|} \hline

Probabilities & \multicolumn{3}{c|}{Axis} \\
\hline
Result & x & y & z \\
\hline
Spin up $\uparrow$ &  &  &  \\
\hline
Spin down $\downarrow $ &  &  &  \\
\hline
\end{tabular}

\vfill

\end{document}