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\begin{document}

\section{Problem Set 2}

\begin{enumerate}
\item
Show that if an operator is self-adjoint in one coordinate system,
it is self-adjoint in all coordinate systems.  (Hint: Transforming
the operator $A$ from one coordinate system to another is done
by the operation $q A q'$, where $q'$ is a unitary matrix.)
\item
Show that $( \psi , A \psi ) = ( A' \psi , \psi )$.
\item
Show that if $A \psi = a \psi$ in one reference frame, where $\psi$
is a vector, $A$ is an operator, and $a$ is a number, it holds in all
reference frames.  $\psi$ is an {\em eigenvector} of $A$, and $a$ is
the associated {\em eigenvalue}.
\item
Find the eigenvalues and eigenvectors of the equation
\begin{displaymath}
	\left( \begin{array}{ccc}
		1.5 & 0.5 & 0 \\
		0.5 & 1.5 & 0 \\
		0 & 0 & 4
	\end{array} \right) \left( \begin{array}{c}
		a \\ b \\ c
	\end{array} \right) = \lambda
	\left( \begin{array}{c}
		a \\ b \\ c
	\end{array} \right) .
\end{displaymath}
Use the eigenvectors to construct the transformation matrix from the
original coordinate system to the principal axis system.  Verify that
the original matrix transforms into principal axis form under this
transformation.  Do these operations both by hand and with RLaB.
(The RLaB routine ``eigs'' finds eigenvalues and eigenvectors ---
do ``help eig''.)
\item
Do the same thing for the matrix
\begin{displaymath}
        \left( \begin{array}{ccc}
                1.5 & 0.5 & 0 \\
                0.5 & 1.5 & 0 \\
                0 & 0 & 2
        \end{array} \right)
\end{displaymath}
(Yes, there is a good reason to redo this with a different $zz$
component!)
\end{enumerate}

\end{document}
