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\begin{document}

\section{The Contributions of Louis de Broglie}

Prince Louis Victor de Broglie, a French nobleman, won the Nobel Prize
in physics for his discovery of the wave nature of electrons.
Actually, his contribution to physics was much broader than this ---
he developed the basic physical ideas that led to the quantum
mechanical treatment of all types of matter.  He is best known for his
relation between particle momentum, $p$, and the corresponding quantum
mechanical wavelength, $\lambda$:
\begin{equation}
	p = h / \lambda ,
\label{momentum}
\end{equation}
where $h$ is Planck's constant, i. e., the same constant used by
Planck and Einstein to relate the energy and frequency of a photon.
Interestingly, de Broglie did all of this for his doctoral
dissertation and thereafter sank into relative obscurity!

\subsection{Momentum-Wavenumber Relation}

Equation (\ref{momentum}) is generally presented without justification
in elementary physics courses.  However, de Broglie showed that this
relationship is an inevitable consequence of special relativity plus
assuming that the Planck-Einstein result,
\begin{equation}
	E = h \nu ,
\label{energy}
\end{equation}
is applicable to material particles as well as photons.

The argument goes as follows: Let us first rephrase equations
(\ref{momentum}) and (\ref{energy}) in terms of angular frequency,
$\omega = 2 \pi \nu$, and wavenumber, $k = 2 \pi / \lambda$:
\begin{equation}
	E = \hbar \omega ~~~~~~ p = \hbar k ,
\end{equation}
where $\hbar = h/2 \pi$.  Now, a plane wave can be represented by
the formula
\begin{equation}
	\psi = \exp [ i ( \bold{k} \cdot \bold{x} - \omega t)] ,
\end{equation}
where $\bold{k}$ is the wave vector.  (Note that $k = | \bold{k} |$.)
In order for the phase of the wave, $\bold{k} \cdot \bold{x} - \omega
t$, to be a relativistically invariant quantity, it must be a
relativistic dot product of two four-vectors, $( \bold{k} , \omega /c
)$ and $( \bold{x} , ct )$.  The second of these is known to be a
four-vector, since it is just a displacement in spacetime.  If the
first is also a four-vector, and if $\omega = E/ \hbar$, then it
follows that this wavenumber-frequency four-vector is just a constant
multiple, $1/ \hbar c$, of the energy-momentum vector.  This leads
immediately to the vector form of the de Broglie relation, $\bold{k} =
\bold{p} / \hbar$.  Thus, de Broglie's momentum-wavelength equation is
not a bolt from the blue, but is a consequence of assuming that the
energy of an ordinary particle is related to the associated wave
frequency by the Planck-Einstein relation.

\subsection{Variable Potential Energy}

In his Nobel Prize lecture, de Broglie also alluded to a method for
computing the wavefunction of a particle subject to a variable
potential energy, $U( \bold{x} )$, by making an analogy with the
behavior of light in a medium of variable refractive index.  For light
with frequency $\omega$ (and hence photon energy $\hbar \omega$), the
equation governing the amplitude, $\psi$, of the light wave is
\begin{equation}
	\nabla^2 \psi_{\omega} + k^2 \psi_{\omega} = 0 .
\label{waves}
\end{equation}
(Note that this is an approximation to electromagnetic wave theory, in
that the polarization of the light is ignored.)  The wavenumber, $k$,
can vary with position, and is related to the wavenumber of the light
in a vacuum, $k_0$, by $k = n k_0$, where $n$ is the refractive index
of the medium.  Note also that $k_0 = \omega / c$.

The trick is in finding the relationship between index of refraction
and potential energy.  This is easily done for the case in which the
potential energy is constant.  In this situation, the non-relativistic
approximation to the relationship between momentum, $p$, and kinetic
energy, $K$, is
\begin{equation}
	p^2 /2m = K = E - mc^2 - U \equiv E' - U ,
\end{equation}
where $E' = E - mc^2$.  Using the relationships between momentum and
wavenumber and energy and frequency, de Broglie found that
\begin{eqnarray}
	k & = & p/ \hbar = [2m (E' - U)/ \hbar^2 ]^{1/2} = ( \omega /c )
		[2mc^2 (E' - U)/ E^2 ]^{1/2} \nonumber \\
        & \approx & ( \omega /c) [2 (E' - U)/mc^2]^{1/2} ,
\label{optical}
\end{eqnarray}
where $\hbar \omega = E$ is approximated by the rest energy $mc^2$
except in terms containing $E - mc^2$.  Thus equation (\ref{optical})
tells us that in the non-relativistic limit, massive particles behave
like light moving through a medium with index of refraction
\begin{equation}
	n = k/k_0 = kc / \omega \approx [(E' - U)/mc^2]^{1/2} .
\end{equation}
Note that in general the index of refraction in this case is very
close to zero, which means that the phase speed of the ``light'' is
much larger than $c$.  However, the group velocity is much less than
$c$, in accord with the non-relativistic assumption.

Assuming that this formula is also valid for variable $U$ leads to a
tentative governing equation for non-relativistic matter waves when
equations (\ref{waves}) and (\ref{optical}) are combined.  Combining
equations (\ref{waves}) and (\ref{optical}), we get
\begin{equation}
	\nabla^2 \psi_{\omega} + [2m (E' - U)/ \hbar^2 ] \psi_{\omega} = 0 .
\label{schrodinger}
\end{equation}
This formula is actually a form of the equation generally credited to
Erwin Schr\"{o}dinger.  The complete wavefunction is obtained by
multiplying by the factor $\exp (-i \omega t)$:
\begin{equation}
	\psi ( \bold{x} , t) =
	\psi_{\omega} ( \bold{x} ) \exp (-i \omega t ) .
\end{equation}

\subsection{A Geometrical Optics Example}

De Broglie's insight shows that quantum mechanical problems involving
the motion of massive particles subject to conservative forces are
analogous to the motion of of photons through a region of variable
refractive index.  The approximation made when the wavelength of light
is much less than the dimensions of the optical system is called
geometrical optics.  In this approximation, wavefronts are replaced by
rays representing the trajectories of photons through the system.  The
analogous approximation to the quantum mechanics of massive particles
is simply classical mechanics.  Thus, if de Broglie's theory is
correct, the trajectory of a photon through a region of varying
refractive index should be the same as the classically computed
trajectory of a particle with the corresponding potential function.
\begin{figure}
\begin{center}
\psfig{figure=example.eps}
\end{center}
\caption{Example of use of de Broglie's index of refraction.  The
optics problem is analogous to the passage of a ray of light from
a medium with high index of refraction into a medium with a lower
value.}
\label{example}
\end{figure}

We can test this idea with a simple example.  Figure \ref{example}
shows a situation in which a ray of light passes from a region
of large refractive index to one with smaller refractive index.
The ray is refracted away from the normal to the interface between
the media according to Snell's law:
\begin{equation}
	\frac{\sin \theta_0}{\sin \theta} = \frac{n}{n_0} .
\end{equation}

In general, equation (\ref{optical}) shows that $n = [2(E' -
U]/mc^2 ]^{1/2}$, so that for the left-hand region in figure
\ref{example} we have $n = n_0 = (2E' /mc^2 )^{1/2}$.  The ratio
therefore takes on the value
\begin{equation}
	\frac{n}{n_0} = \left( \frac{E' - U}{E'} \right)^{1/2} .
\label{nratio}
\end{equation}
However, according to non-relativistic classical mechanics, $K = E' =
(p_x^2 + p_z^2 )/2m$ in the region to the left of the interface, while
$K' = E' - U = ({p'}_x^2 + p_z^2 )/2m$ to the right, where the $p$s
represent momentum components with $x$ to the right and $z$ upward.
The $z$-component of momentum is conserved because the force $-\nabla
U$ has only an $x$ component.  Thus only the $x$-component of momentum
changes when the particle crosses the interface.

The angles can be related to the components of the momentum vector:
$\sin \theta_0 = p_z / (p_x^2 + p_z^2 )^{1/2}$ and $\sin \theta = p_z
/ ({p'}_x^2 + p_z^2 )^{1/2}$.  Thus,
\begin{equation}
	\frac{\sin \theta_0}{\sin \theta} =
	\left( \frac{{p'}_x^2 + p_z^2}{p_x^2 + p_z^2} \right)^{1/2} =
	\left( \frac{E' - U}{E'} \right)^{1/2} ,
\end{equation}
which is in accord with the result derived from geometrical optics,
equation (\ref{nratio}).  Thus, the equivalence of geometrical optics
and classical mechanics is established at least for this example.

\end{document}
