# Potentials and curl

1. Oct 23, 2008

### sirchasm

I'm trying to get a picture of magnetic potential, as in how to relate it to spin precession.
(So have I got the right picture so far?)

Classically, B is measured as the derivative of curl (which is a circulation integral) "around" a conducting current (that is, perpendicular to the direction of current I).
Gauss' Law says the field is divergence-free, which means it circulates linearly, and the integral is just BL, for any circular integral along a circumference L, of a potential around say, a 1-d current in a wire.

Einstein and de Haas tried to measure precessional forces to explain magnets, by envisaging electrons as being in orbits and precessing, as a gyroscope does in a gravitational potential. It didn't quite explain magnetism because it didn't allow for Pauli's exclusion principle that 'admits' two electrons, which share an orbital; magnetic forces are in fact related to electron's spin angular momentum, not their orbital momentum. Or electrons rotate "in place", rather than orbiting like gyroscopes.

Aharonov-Bohm phase precession is due to a magnetic potential in a curl-free region, which rotates the spin precession angles in a divided beam of electrons.
It looks like a bump for 1/2 the beam, and like a dip for the other 1/2. One spin phase shifts up or 'squeezes' over the dip, the other shifts down or 'expands' over the bump in potential. The phase-difference is preserved, as momentum carries the matter-wave to a detector.

So it's about how to relate the geometry of a magnetic field (potential) to the geometry of spin in electrons.
Is potential just equal to the precession angle, since a change in this angle, will change the local potential gradient?

It's a question (I think), of seeing the geometry and topology involved. The algebra is the really tricky bit with complex quantum spaces with a dimension of $$\mathbb C^{2n}$$. Anyways, there's this somewhat dated article in a '81 SciAm about the A-B geometry and neutron spin-precession. Quite an interesting geometric picture; they relate geodesics (over an inclined conic surface) to parallel transport, then what a "path-lifting" does, and so on.

You can relate the observed patterns of interference to the topology and geometry of spin precession. Spin is a 'component' of the Schrodinger wavefunction(?), and charge is too(?), whereas position and momentum are another component, or IOW the wavefunction is a "mixed wave" of spin, charge, and mass wave components? There's a major difference between the A-B experiment and neutron interferometry, wrt to the magnetic field applied, since the neutrons cross classical field-lines, or do not travel through a curl-free region, but see a field that rotates the precessional angle, over time, whereas the electrons see just a potential and preserve a single phase shift?

Have I left something out of this picture?