Friday, December 21, 2012

Lamb and Scully on Stern Gerlach

About a year ago I posted some articles about how the traditional explanation of the Stern Gerlach experiment is all wrong. Everyone says you shoot a beam of silver atoms through the apparatus and it splits into two beams. So if you put a screen up and collect the atoms, you get two silver dots on the screen.

There is no such experiment. It can't be done. I explained this last year, and drew a very nice picture of exactly what the beam does. You don't get two dots on the screen; you get a donut that looks like this:

Actually, that's the picture for a polarized where the silver atoms are pre-selected so they're all spin-up. I even worked out the wave equation, expressed in terms of the angular dependence of phase. It looks like this:

At least that's what I figured out. I haven't found any source to confirm my result...until today. I was reading the Wikipedia article on the Stern Gerlach experiment and it included a link to this article by Lamb, Scully and Barut.   (It's a Springer article and I was pretty surprised when it opened up without me having to pay for it.) The article is from 1987 and the authors make pretty much the same point I was making last year, and they also come up with an equation for the wave function. Here is the equation they give:

I don't know about you, but I can't exactly figure out what it all means.The beta's which they are summing over seem to be the spinor components, so I think they are treating the case of the unpolarized beam.  It's quite possible that this formula is saying exactly the same thing as my much simpler formula, but I just can't tell. The quadratic in the argument of the exponential almost suggests to me that they're trying to actually follow the wave function of the silver atom as it passes through the lengthwise magnetic field, whereas my function is essentially the far-field outcome. But it's all just a little over my head.

So I'd be interested if anyone can tell me: based on the result from Lamb and Scully, is my formula right or wrong?

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