
\subsection{\bf The Laplacian as a (Matrix) Operator}

\begin{slide}

\begin{itemize}
\item $\nabla^2 = \frac{d^2}{dx^2} = -L\,L$
\vspace{12pt}

\item $\left(\nabla^2 \right)^\dagger = -L^\dagger L^\dagger = -L\,L = 
\nabla^2$
\vspace{12pt}

\item Note: $(A\,B)^\dagger = B^\dagger A^\dagger$ in general
\vspace{12pt}

\item Thus, $\nabla^2$ ($\frac{d^2}{dx^2}$ in one dimension) is
Hermitean (symmetric) operator
\end{itemize}

\end{slide}

