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\begin{document}

\section{Time Evolution and the Energy}

If the possible values of frequency for some particle are $\omega_j$,
$j = 1,2,3, \ldots$, and the amplitude to have frequency $\omega_j$ is
$A_j$, then the expectation value of frequency
is
\begin{equation}
	< \omega > = \sum_j A_j^{*} \omega_j A_j = (A, \omega A) .
\label{expect}
\end{equation}
The position space representation of the particle state represented
by the values $A_i$ must simply be given by
\begin{equation}
	\psi (x_m ) = \psi_m = \sum_j {q'}_{mj} A_j ,
\end{equation}
or
\begin{equation}
	\psi = q' A
\label{space}
\end{equation}
where ${q'}_{mj}$ is the matrix representing the unitary
transformation from the energy representation of the state vector to
the position representation.  The inverse transformation is just
\begin{equation}
	A = q \psi .
\label{initial}
\end{equation}
Proceeding as with the momentum operator, we can therefore write
\begin{equation}
	< \omega > = (q \psi , \omega q \psi ) =
		( \psi , q' \omega q \psi ) ,
\end{equation}
where $q' \omega q$ is the frequency operator in position space.

So far we have not asked how state vectors in {\em any} representation
vary in time.  However, we know from the work of Planck, Einstein, and
de Broglie that the amplitude $A_j$ for a particle with definite
frequency $\omega_j$ has a time dependence $\exp (-i \omega_j t)$.  We
can therefore construct the time-dependent state vector resolved in
position space by assuming that $A_j \propto \exp (-i \omega_j t)$:
\begin{equation}
	\psi (t) = q' A(t) .
\label{evolution}
\end{equation}

Notice that
\begin{equation}
	i \frac{\partial \psi (t)}{\partial t} =
	q' i \frac{\partial A}{\partial t} =
	q' \omega A =
	( q' \omega q ) q' A = ( q' \omega q ) \psi .
\label{timederiv}
\end{equation}
Thus, the frequency operator in position space is simply $i$ times the
time derivative.\footnote{More properly, to keep our analogy with
finite matrices, we should write this representation of the frequency
operator as $i \delta (t - t' ) ( \partial / \partial t' )$ and the
left side of equation \ref{timederiv} as $i \int \delta (t - t' ) [
\partial \psi (t' ) / \partial t' ] dt'$.}

Equation (\ref{timederiv}) is generally multiplied by $\hbar$, turning
the frequency into energy.  The operator $\hbar q' \omega q'$ is
called the {\em Hamiltonian}, $H$, and it is just the operator whose
eigenvalues are the possible energies of the system.  The resulting
system is just a generalization of Schr\"{o}dinger's equation:
\begin{equation}
	i \hbar \frac{\partial \psi}{\partial t} = H \psi .
\label{schrodinger}
\end{equation}
This may be seen to be true for a single non-relativistic particle if
the Hamiltonian, or energy operator, is related to the position and
momentum operators as in classical physics.  In particular, for a
non-relativistic particle of mass m subject to the potential $U(x)$,
\begin{equation}
	H = \frac{p^2}{2m} + U(x) =
	-\frac{\hbar^2}{ 2m} \frac{ \partial^2}{ \partial x^2} + U(x) ,
\end{equation}
and this equation becomes
\begin{equation}
	i \hbar \frac{\partial \psi}{\partial t} =
	- \frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x} +
		U(x) \psi .
\end{equation}

The mathematical structure underlying quantum mechanics is now almost
complete.  The goal in quantum mechanics is to obtain the Hamiltonian
operator for the physical system of interest in some representation.
Then the Schr\"{o}dinger equation (\ref{schrodinger}) is either solved
directly or the eigenvectors, $\psi_j (x)$, and eigenvalues, $E_j =
\hbar \omega_j$, of the Hamiltonian are found by some other means.  In
the latter case the evolution of the state vector in the original
representation is obtained from equation (\ref{evolution}), with $A$
at time $t = 0$ being obtained from equation (\ref{initial}).  The
advantage of this approach over de Broglie's and Schr\"{o}dinger's is
that it applies to {\em any} system for which the Hamiltonian can be
found --- not just the case of a single non-relativistic particle.

\end{document}
