The Quadrupole Field: Tidal Forces and Particle Beams

People have an awful time eplaining the tides. Everyone knows the moon attracts the oceans, making them bulge out on one side: but how do you explain the second bulge on the wrong side of the earth? The tides go in and out twice a day, not once. How do you explain it?

The mistake people make is to think that it’s the pull of the moon that causes the tides. It is and it isn’t. There is absolutley no tidal force in a steady gravitational field. The tidal force occurs because the moon’s gravity is not a constant over the volume of the earth. It get’s weaker as you get farther. It’s the variational component of the steady field which is responsible for the tides. You can draw a picture of it easily:

The big grey arrow represents the average magnetic field in the vicinity of the earth, and it has absolutely no influence on the tides. The four little black arrows, representing the deviation from the steady field, are entirely responsible for the bulges in the ocean. You can see in the picture that the distortion of the ocean is exactly in line with the four small arrows. These arrows represent what is called the “quadrupole component” of the field, and they cause the tides.

It is very significant to notice that the up/down arrows are the same strength as the left-right arrows…at least I’ve drawn them that way. In real life, they are exactly half strength…because there are two more off-axis arrows in the actual field, representing the component in and out of the page. The TOTAL strength of the four off-axis arrows is equal to the strenght of the on-axis arrows. This is not an accident, and it’s not something that applies only to gravitational fields. It’s a basic property of all electric and magnetic fields that is also known as Gauss’s Law: flux in is equal to flux out.

I bring this up because it applies in a huge way to the Stern Gerlach experiment. Everyone, and I mean everyone explains Stern-Gerlach by saying the field is stronger near the pointy magnet and weaker near the flat magnet. Yes it is, but what about the off-axis component? It’s just like the moon’s gravity! In fact, it’s even more so, because for the moon, we have three-dimensional geometry, so the off-axis field lines are weaker by a factor of 2; but in the two-dimensional geometry of the long pointed magnets, the off-axis component (which everyone  ignores!) is exactly as strong as the on-axis component. Just as I’ve drawn in the picture.

And just as the steady-state component of the moon’s gravitational field has absolutely no effect on the tides, why should the steady-state component of the magnetic field have any effect on the silver atoms? Everything that happens in Stern-Gerlach has to be explainable on the basis of the quadrupole component of the field. Why should it be different from the tides?

But if this is true, the implications are huge. Everybody says that the beam splits in two. Even Feynman talks about splitting the beam, and then taking one component of the split beam and putting it through a second Stern-Gerlach magnet rotated through 90 degrees; he says then it will split in two again. He even goes so far as to acknowledge that the experiment has never been done quite this way, but he doesn’t doubt that it is theoretically possible.

And why shouldn’t it be possible? Everyone knows the beam splits in two. But does it??? In the original Stern-Gerlach experiment, there was no pencil-shaped beam: the beam came out of an oven through a slit, not a pinhole, and the orientation of the slit was perpendicular to the axis of the magnet. The silver atoms came out in a fan-shape, and spread into two lines on the detection plate…not two dots, but two lines. If I am correct, and I can’t see why I wouldn’t be, there is no way to split a pencil beam into two dots. Because the quadrupole field has just as much splitting force in the left-right direction as it does in the up-down direction.

That's why you can't expect the beam to split into two dots. If you haven't been following my blog regularly, you might have missed my post from last month when I calculated the spreading of an actcual beam through the quadrupole field. Instead of two dots, you get a donut that looks like this:

That's the pattern for a beam that goes in with it's spins polarized in the up direction. For an unpolarized (random) beam, you just get a simple donut pattern.

There is no physical way to construct a magnetic field that gets stronger from top to bottom without at the same time have it show just as much variation in the left-right direction. So it is impossible to split a pencil-shaped beam into two dots. Yes, you can split a fan-shaped beam into two lines, but that is not at all the same thing. In particular, all of Feynmann’s thought-experiments that he works through in Vol. 3 Chapter 5…are based on a non-physical misinterpretation of the apparatus. Feynmann got it wrong. Oh, it’s all right for the point he’s making about adding amplitudes and transformation of basis states…all that is immaculate. But the actual experiment does not exist.