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\centerline{\textbf{The Finite Square Well}}
\bigskip


This graph (solid black line) represents the potential energy of a particle at any position.  This is known as the "finite square well" problem, and the potential function is a piecewise function

$$
V(x) = 
\begin{cases}
V_{0} & x < -a \\
0 & -a < x < a \\
V_{0} & x > a \\
\end{cases}
$$

The dashed line represents the total energy $E$ of the particle.  We don't know the value(s) of $E$ yet; we have to find them.  We also don't yet know the form of the wave function $\phi (x)$.  We have to find the correct form.  For this piecewise potential function, the wave function has a different form in each of the regions.  Solve the eigenvalue equation $\hat{H}\phi (x) = E\phi (x)$ in the region assigned to your group.  Afterwards, we will have a group discussion to decide how to make sure that the total wave function over the entire space is appropriately continuous and otherwise well-behaved.

\begin{enumerate}
\item	Region 1: $x < -a$
\item	Region 2: $-a < x < a$
\item	Region 3: $x > a$
\end{enumerate}

Your group will discuss one of the regions: if you finish early, move on to another, swapping roles of taskmaster, cynic, \& recorder.  Consider whether or not your answer would be different if $E > V_{0}$. 

\begin{figure}[h]
	\centering
		\includegraphics[width=5 in ]{wvfinitewellfig1.jpg}
	\label{fig:wvfinitewellfig1}
\end{figure}


\vfill
\leftline{\textit{by Janet Tate}}
\leftline{\copyright DATE Janet Tate}
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