Thursday, December 15, 2011

Stern-Gerlach without the Quantum Leap

The Stern-Gerlach experiment is often put forward as a showcase example of wave function collapse. The silver atom enters the magnetic field with a random spin, and suddenly it has to make a choice: spin up or spin down? Depending on the answer, it takes one of two paths to the screen where it is detected.
This widely believed description is in fact a deliberate, almost criminal mystification of the process. Last week I said I would show how to understand the Stern Gerlach experiment as the normal time-evolution of the Schroedinger wave function, and today I’m going to get started.
The simplest case to deal with is when you have a beam of atoms prepared in the spin-up state: that is, their spins are already aligned with the magnet. We want to show that the beam curves upwards. Of course, it does so classically: the tiny atomic magnet is simply drawn to the stronger pole. The problem with this analyis is that if the magnet is tilted, it should still be drawn in the same direction, only not so strongly. That’s not what happens.

We need a way to analyze it which not only gives the right curvature in the case of aligned spin, but also gives us the bifurcation of paths for the case of arbitrary spin. We have to analyze the wave function of the beam. And it is especially helpful to let the beam have some finite width, as we will see in the picture below:

You can see that like any wave in quantum mechanics, the phase of the wave precesses faster when it has a higher energy…in this case, the portion of the beam near the bottom. (EDIT: This is really wrong. I didn't realize until over a year after I posted this that I didn't have the right explanation for the bending of an electron wave in a potential field. I finally worked it out and you can read the correct explanation HERE. But even so, my picture is still correct!) So the wavefronts tend to get ahead of the rest of the beam. That has the effect of tilting the imaginary suface which represents a constant phase. You can see that the effect of this is to bend the beam upwards. You would draw exactly the same diagram to show the path of an electron bending around a charged object.
Similarly, if the silver atoms were aligned the opposite way, the phase would advance faster at the top of the beam, and it would bend downwards.
But what if the spin is pointing sideways? The truly magical thing about spin is that you can represent a sideways spin as the composition of up and down spins, just as in electromagnetics you can represent a 45 degree polarization as the composition of vertical and horizontal. You just break up the spin into its up and down components, and analyze them separately. When you put the results together, you see that the beam path divides in two when it passes through the magnets. That’s what happens.
However, I don’t find this mathematical answer to be completely satisfying. We know, for example, that if an atom enters the field with its magnetic axis tilted sideways, then classically it ought to precess. That is, the direction of spin should rotate within the plane perpendicular to the external field. Does it also do this quantum mechanically? And if so, how can we reconcile this with the path dividing in two when the beam leaves the magnet?
It turns out we can indeed analyze it this way, and although the picture looks quite different, it turns out to have the same result. It’s a matter of your choice of “basis states”, and it’s an exercise well worth carrying through in order to develop your intuitions of how these things work. But before we go there I want to return to the picture we already drew and put a slightly different perspective on it.
I’ve drawn the same picture but I’ve added an imaginary surface representing the place where the magnetic field ends. I’ve drawn it supposedly parallel to the orientation of the original phase fronts; so the waves are tilted slightly with respect to this plane as they leave the magnetic field.

If you carefully trace the wavefronts in that imaginary plane, you’ll see that they’re something like I’ve shown with the orange line. It’s not quite right because these are actually exponential waves, not sine waves, so the amplitude never goes to zero. But it’s kind of traditional to draw them as sine waves anyhow because that’s the best we can do.
The thing to notice is that if we “unfreeze” the static picture and imagine what those wavefronts are doing in real time, they are actually zipping along from south to north at quite a good pace, quite possibly even faster than the speed of light. There is nothing unusual about this: it’s a very normal circumstance and it happens all the time in physics. Remember these aren’t real physical waves carrying energy: they’re just mathematical points along an imaginary surface.
In fact, it’s exactly the same picture you’d draw if you were trying to explain reflection from a mirror. The light waves come in from the left and strike the mirror at a glancing angle. The fields set up oscillating charges in the surface of the mirror. If we trace the charge oscillations, we find that they are just the waves shown by the orange line in the picture: they move from south to north and the “wave” definitely travels faster than the speed of light. It’s an obvious consequence of the glancing angle.
It’s this rushing wave of charge oscillations in the mirror that generates the reflection. There’s something exotic sounding called Cerenkov radiation and it has to do with the waves generated by a source travelling faster than light. It sounds like stuff that you get in cyclotrons or whatever, but in fact it’s a very ordinary thing. The current waves in the mirror travel faster than light, and as a result they generate wave systems peeling off at an angle, represented by the blue arrows. The dashed arrow is the reflected wave.
What about the solid arrow? That’s the wave that cancels out the incident wave! Otherwise the incoming wave would just pass right through the mirror and keep on going. The waves of charge in the mirror do two jobs: they cancel out the incoming wave, and they generate the reflected wave.
This is the exactly the same methodology we are going to use when we analyze the Stern Gerlach experiment. I’m going to save that for another day.

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