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\centerline{\textbf{Operators \& Functions}}
\bigskip

\bigskip

For each of the following operators:
\begin{itemize}
\item Test each function to see if it is an eigenfunction of the operator.
\item If it is, what is the eigenvalue?
\item If it is not, can you write it as a superposition of functions that are eigenfunctions of
that operator?
\end{itemize}

$\begin{array}{l c l}
1.\hs  \hat{p} = -i \hbar \frac{d}{dx} & &

\psi_{1}(x)=Ae^{-ikx} \\
&&\psi_{2}(x)=Ae^{+ikx} \\
&&\psi_{3}(x) = A \sin{(kx)}
\\
&&\\
2. \hs \hat{H} = - \frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}} && 
\psi_{1}(x)=Ae^{-i\frac{p}{\hbar}x}\\
&&\psi_{2}(x)=Ae^{+i\frac{p}{\hbar}x}\\ 
&&\psi_{3}(x) = A \sin{(\frac{p}{\hbar}x)}\\ 
&&\\
3. \hs \hat{H} = - \frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}} &&
\psi_{1}(x)=A \sin{(kx)} \; \; \\
&& \psi_{2}(x)=A \cos{(kx)} \; \; \\
&& \psi_{3}(x) = Ae^{ikx}\\
&&\\
4. \hs \hat{S}_{z} \rightarrow  \left(\begin{array}{cc} 
\frac{\hbar}{2} & 0 \\
0 & -\frac{\hbar}{2} \\ \end{array}\right) &&
\vert \psi_{1} \rangle = \left(\begin{array}{c} 1 \\ 0 \\ \end{array}\right) \; \; \\
&&\\
&& \vert \psi_{2} \rangle = \left(\begin{array}{c} 0 \\ 1 \\ \end{array}\right) \; \; \\ 
&&\\
&& \vert \psi_{3} \rangle = \left(\begin{array}{c} 1 \\ 1 \\ \end{array}\right)\\



\end{array}$

Are superpositions of eigenfunctions of an operator themselves eigenfunctions of the same operator?

\vfill
\leftline{\textit{by Janet Tate}}
\leftline{\textit{updated by Liz Gire, 2020}}
\leftline{\copyright 2003 Janet Tate}
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