Friday, December 16, 2011

How to Avoid Math by Using Physics

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.

Actually, when I drew this picture, it wasn’t meant to represent the case of the mirror. I was really doing the  analysis of a particle trajectory bending as it passes through an external field. The particle could have easily been an electron passing through a charged capacitor, but again that’s wasn’t the idea. The diagram was actually drawn to show a beam of silver atoms passing through an inhomogenous magnetic field: the Stern Gerlach experiment. But either way, the drawing is the same! The “particles” are represented by a wave function, and the phase of the wave function changes more rapidly where the energy is higher. The advancing phase thereby imparts a curvature to the path. That’s how it works.

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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