It’s taken a while but I’ve laid out all the machinery for
analyzing the case of the silver atom entering the magnetic field tilted
sideways. Here is how it goes.

1. The atoms precess in the magnetic field. The ones higher
up precess faster.

2. Because of the
differential rates of precession, when they leave the magnetic field the spin
orientations form aa corkscrew pattern across the cross-section of the beam.
3. Now break the corkscrew into up and down components and
analyze the output as a diffraction pattern.

For purposes of calcualtion we can take the precession to be
zero in the center of the beam. From basic spinor algebra, we know that the
spin-up component will vary as the cosine of the distance from the center;
likewise, the spin-down component will vary as the sine. Here is a map of the
spin components across the cross-section:Of course, I’ve taken my “up/down” orientation to be aligned with the spin of the atoms. So what I’m calling “spin-up”, the original orientation of the atoms, is actually spin-sideways to the magnetic field. It’s

*easy*to analzye if I take my spin basis to line up with the magnet…I did that last week. The interesting thing is how it works if I take the alternate spin basis, and that’s what we’re about to see.

From everything we’ve gone over in the last week, it should now
be obvious that the beam splits in two. The spin-up component splits in two,
and the spin-down component splits in two. From the incoming beam, we get two
outgoing beams.

What can we say about the spins of the two outgoing beams?
They appear to be made up of equal portions of spin-up and spin down. But look
carefully at the relative phases of the two components. They are out by 90
degrees, and they are out in opposite directions! In one beam the spin-down is
90 degrees *behind*the spin-up, and in the other beam it is 90 degrees

*ahead.*

I’m not going to go back and derive all of spinor algebra
from scratch, but if you know anything about spinors you can see that these two
beams have their spins alignede in opposite directions. In fact, one is spin-up

The next problem on my agenda is the one proposed five years
ago by Brigham Young grad student Jared Rees Stenson: the splitting of the beam
in a pure quadrupole field. I wonder how that’s going to look.*relative to the Stern-Gerlach magnet*and the other is spin-down. It’s the same result we got when we analzyed it in the spin-basis parallel to the Stern-Gerlach field, but now you can actually see how it works with precesssion and recombination.
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