Electron Stern-Gerlach Magnets

The following is part of the abstract of our paper in American Journal of Physics 70, 325 (2002).:

One of the cornerstones of quantum mechanics is the Stern-Gerlach effect. An unpolarized beam of silver atoms is passed through a strong magnetic field gradient and splits into two polarized beams. This effect is one of the main motivations to postulate that electrons have spin, in particular spin-1/2. Particles carrying an angular momentum, J, split into 2J+1 parts; for integer values of J there will be an odd number of beams. The existence of half-integer spin was postulated to explain the observation of splitting into an even number of beams. Because the total angular momentum of silver atoms is half-valued (which can be traced to the electron spin), a beam of silver atoms splits into two parts. This is, of course, the result of the famous Stern-Gerlach experiment 1. The dynamical picture of the Stern-Gerlach effect involves the interaction between the magnetic moment associated with a charged spin-1/2 particle and the magnetic field gradient. A spin 1/2 particle with its projection of spin in the positive/negative z-direction (spin-up/spin-down) will experience a deflection. We can think of this as a small magnet passing through and being deflected by a magnetic field gradient. The quantum mechanical rules for angular momentum allow only two possible orientations (up and down) for "the small magnet", in contrast to all possible orientations allowed classically. This explains why the Stern-Gerlach effect is quantum mechanical.

From this description one might expect that a beam of free electrons would also split after passing through a Stern-Gerlach magnet. However, Mott, Bohr, and Pauli have posed that this is impossible due to the blurring effect of Lorentz forces on a finite-size beam 2,3. A very narrow beam would not suffer from spatially varying Lorentz forces, but would be so badly diffraction limited that it would no longer be a beam. Even alternative approaches designed to overcome the Lorentz forces 4 have been rejected by Pauli. One of these, the longitudinal Stern-Gerlach experiment, originally proposed by Brillouin, has been reinstated by us: Phys. Rev. Lett. 79, 4517 (1997). In the longitudinal Stern-Gerlach magnet, the magnetic field gradient is aligned with the electron beam, and spin "forward" and spin "backward" electrons passing through the magnet are separated along the propagation direction (figure 1b). To analyze this experimental situation, one can not resort to the usual semi-classical dynamical picture where the electron motion is treated classically and the spin quantum mechanically. The reason is that to overcome the blurring Lorentz forces, we would like to approach the diffraction limit without losing the beam. This means that the electron motion should also be treated quantum mechanically. We recently analyzed this situation with the Schrödinger equation and obtained the promising result that the spin-splitting is much better than our semi-classical analysis indicated. Phys. Rev. Lett. (2001)