Stern Gerlach with a Quadrupole Field

After I wrote my last article, I happened across an article with a different perspective on the Stern Gerlach experiment. It was actually a Master’s Thesis from one Jared Rees Stenson, a student at Brigham Young University, written in 2005.

This Jared makes an interesting point. Everyone talks about the Stern Gerlach experiment as though you shoot a thin beam between two magnets, and it divides into two: so you get two splotches of silver on the glass plate. I pointed out in my last article that you really need to consider the thickness of the beam, as though it were the size of a pencil. Jared makes a slightly different point: he says that Stern and Gerlach used a collimated slit  for their source, so the silver beam was really more fan-shaped than ray-like. And that the pattern on the glass plate was more of an ellipse than anything else:

Jared then goes on to doubt that if you actually had a pencil-thin beam, you would in fact get the two perfect dots that people like to talk about. And I think he may be right. He points out that classically, you really can’t create a magnetic field which gets stronger from top to bottom without also having it fan out. It’s basically Gauss’s Law applied to magnetostatics, and nobody really accounts for this when they analyze the “ideal” Stern-Gerlach system.

But what I like most about his thesis was where he suggested a different way of setting up the magnetic field, so as to eliminate the DC component. He arranges four wires to create a perfect quadrupole field, like so:

He then asks the question: if you shoot a pencil beam of silver atoms through this magnetic field, what pattern do you get on the glass plate? If you think about it, it's a funny question.

It turns out Jared is some kind of monster mathematician, and he does a bunch of stuff that I don't really follow to end up with the nice result that the pencil beam becomes a donut on the glass plate.

Now that sounds right to me if you start out with a random, unpolarized beam. But it occurs to me: what if you have a polarized beam...one that's already been selected for spin in one direction by a conventional Stern-Gerlach filter? What pattern do you get in that case?

It's a question that I have to ask myself almost as a matter of put-up-or-shut-up: in my last post, I said I could analyze the conventional S-G experiment by standard wave theory, and if I'm so smart, why can't I do it for this one too? It looks like a fun question to work on and I have some idea how it ought to come out. So give me a day or two and let's see what I come up with...