When I left off yesterday, I was talking about how you can
analyze reflection from a mirror by looking at waves of current in the metal
surface. These current waves flow in response to the incoming electric field;
but if you ignore the driving field, they just look like waves of current
flowing in the metal. It’s an odd fact that these waves seem to skim over the
surface of the metal at speeds much greater than the speed of light; but
there’s nothing physically mysterious about this. It’s just a geometrical
consequence of the glancing angle with which the driving waves impinge upon the
surface.

There are of course physical consequences to these current
waves: they must in turn create electromagnetic radiation. It’s a true
consequence of electromagnetic theory that a source travelling faster than the
speed of light must radiate, and the simple mirror is a perfect example of this
phenomenon. It is a bit of work to actually calculate it from scratch, but the
result of the calculation can be easily guessed from physical considerations. There
are two waves generated, shown in my sketch by the solid and the dashed arrows.
The dashed arrow represents the reflected wave, and the solid arrow represents
the wave which perfectly cancels the incoming wave. That’s why you can’t shine
a beam of light through a mirror. The currents flowing in the metal surface
generate a wave which is exactly equal in magnitude to the incoming wave, but
opposite in phase.

Unfortunately it’s not always that easy to follow the wave
trajectory. It’s easy in this case because we can identify surfaces of constant
phase and magnitude, and it’s easy to see that the power in the wave must flow
perpendicular to these surfaces. But it’s not always easy to identify such
surfaces; indeed, they don’t always exist. So we need to develop other techniques.
Which brings us to the present case.

I’ve already shown what the waves do if you throw a mirror
into the picture. But the funny thing is they do pretty much the same thing *without*the mirror! Oh, it’s not really the same at all once you get right down to it. But the case of a mirror gives us really good insight into the mathematical form of the actual solution.

The main idea is that every point on a wave acts as a source
for spherically outgoing waves. Every point. The overall sum of all these infinitely
many spherical sources gives you the resulting wave. In the case of the
ordinary plane wave propagating in the forward direction, all those little
spherical wavelets cancel out everywher except in the normal direction of
propagation. The backwards-going wavelets cancel out. The sideways-going
wavelets cancel out. Only the forward-going waves are constructively
reinforced.

In the case of the Stern-Gerlach experiment, instead of
looking for surfaces of constant phase, we can just throw an arbitrary imaginary
surface into the picture and ask what the wave looks like as it passes through
that surface. Since in this case it is just slightly oblique to the alignment
of the phasefronts, it works just like the mirror. The Schroedinger wave seems
to be skimming along this surface at a speed much greater than the actual
particle velocity, or even the velocity of light. It’s just like the wavefronts
on the surface of the mirror, and there’s nothing unusual about it.
The difference is that in the mirror situation, these were
actual currents flowing in response to the driving field. In this case there
are no currents flowing because there is no surface for the to flow within: it’s
just the mathematical value of the wavefunction that we’re considering. But
remember how we said we would analyze it: we treat every point on the wave
function as an independent source of spherically outgoing waves, and calculate
the sum of all those sources.

It sounds like a monstrous mathematical task, and in fact it
is. It’s almost impossible to do it. But fortunately we don’t have to. Nature
has already shown us the solution in the case of the mirror. The solution
consists of two outgoing waves shown by the dashed arrow and the solid arrow.
The difference is that in the case of the mirror, the dashed
arrow represented the actual solution, namely the reflected wave; and the solid
arrow represented the imaginary wave that serves to cancel out the incoming
power. But the beam of silver atoms is a different story.

You still get the dashed wave and you still get the solid
wave. But you get much more than that. I drew an arbitray plane in the middle
of the picture. I could have drawn a similar plain one micrometer to the left,
or one micrometer to the right. Each one of those planes would show similar
solutions to the wave equation, just displaced by the corresponding distance.
The result of adding all these plane source solutions is
easy to guess: in the backwards direction, shown by the dashed arrows, the
sources are cumulatively out of phase so they end up completely cancelling out.
A freely travelling wave does not spontaneoulsy generate a backward-moving
reflection.

But the forward-travelling directions reinforce each other.
The wave moves forward, and its direction is shown by the solid arrow. We can
be lazy and pretend that we’ve solved the for the resultant wave just by
looking at the effect of the sources in our single, arbitrarily selected plane,
but in honest truth that only gives the barest whisper of the shape of the
solution. The true solution is identical in form except it’s reinforced by the
cumulative effect of every other imaginary plane.
What is the point of all this? Simply that for normally
propagating waves, we can identify the general form of the complete trajectory
by evaluating the wave function over a single, planar surface within the path
of the wave. Ordinarily this is not a tremendous benefit, because we still have
to do a monstrous amount of calculus to integrate over all the sources. But
sometimes, as in the present case, we can identify the form of the wave with a
physical situation where we already know the answer. That’s what we’re going to
do when we take up the next topic: the case of the sideways-polarized beam in
the Stern Gerlach experiment.

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