Xjyuk

Many questions have been asked on this site about the Stern-Gerlach experiment, but as far as I can tell this one hasn't. Does the following classical explanation of the SG experiment work?

Model electrons as a finite-sized hollow sphere of rotating charge $e$. Immediately after entering the SG device, the dipole moment of the electron quickly reorients itself to align with the magnetic field of the device. This is what I would expect to happen if a small bar magnet were placed in a region of high magnetic field gradient. Suppose this happens before the electron has traveled 1% of the distance of the device. Then, for the remaining 99% of the distance in the SG device, the electron is completely oriented either "spin-up" or "spin-down," so the binary "all-or-nothing" measurement is naturally predicted by this classical picture.

This would also work to explain sending the beam through multiple differently oriented SG devices, since the previous spin orientation of the electron is completely altered (very quickly) each time the electron enters a new device.

It's a decent question, as many people probably thought the same when the Stern-Gerlach experiment was first released. But there are many hurdles if you try to explain it classically. To get you started:

• For spin $1$ particles, you get three spots and not two.


• There is no mechanism given to dissipate the energy, to make it stay.


• It is not simple to make a classical model where both spin up and spin down are energetically favored. In a straightforward model, if spin up has the highest energy, spin down has the lowest.


• Microscopic classical models of spin generically get the spin wrong by a factor of $2$ and, given experimental data available at the time of the Stern-Gerlach experiment, require parts of the electron to be moving faster than light.


• If you pass electrons through a vertical SG apparatus and select the spin up ones, then pass it through a horizontal SG apparatus, recombine the beams, and pass the result through a vertical SG apparatus, all of them will be spin up. This makes no sense in a model where a horizontal SG apparatus just rotates the spin to horizontal.

This is also ignoring the issue that permanent magnetic moments can't even exist in classical mechanics.

Modeling the electron as a finite-sized, spinning, charged spherical shell doesn’t work. Among a variety of problems, this model predicts the wrong value for the electron’s magnetic moment.

Consider a spherical shell of mass $m$, charge $e$, and radius $a$, spinning at angular velocity $omega$. Its moment of inertia is

$$I=frac23ma^2$$

and its magnetic moment (in Gaussian units) is

$$mu=fracea^23comega.$$

(Source: http://www.physics.princeton.edu/~mcdonald/examples/rotatingshell.pdf)

The angular momentum is

$$L=Iomega$$

and we know that for an electron this is $hbar/2$.

We can then express the magnetic moment as

$$mu=fracehbar2mc.$$

But the magnetic moment is measured to be slightly greater than twice this value.

There has been a recent demonstration of a Stern-Gerlach atom interferometer with high accuracy control of magnetic gradient on an atom chip (https://arxiv.org/pdf/1801.02708.pdf). Any attempt to picture the magnetic moment as a classical dipole pointing in a certain direction will fail to explain the interference pattern observed (see Figure 2 in the paper).

First, in the Stern-Gerlach experiment, a beam of silver atoms was used, and not a beam of electrons. Secondly, the interpretation of data at that time (1922) based on the classical equations of motion. It is assumed that the silver atom has a magnetic moment on which the force acts

$$vec F=nabla (vec mu.vec B)……(1)$$

The role of quantum mechanics is reduced to the statement that the magnetic moment in equation (1) is quantized in proportion to the angular momentum or spin. In this form, the theory of the scattering of atoms in a non-uniform magnetic field was used to solve practical problems related to the determination of the magnetic moment of atoms using the Stern-Gerlach method down to the present time. As it is known, Stern and Gerlach obtained the following figure of silver atoms (on the left without a magnetic field, on the right in a magnetic field)

The question is, can we get the right figure using equation (1) with $mu =pm mu_B$? I simulated the scattering of silver atoms in a magnetic field with a gradient as in the Stern-Gerlach experiment. All parameters of the gap and velocity of atoms were as in experiment. The result is shown in Fig. 2, where a) magnetic field lines and gap projection (red rectangle); b)magnetic field gradient (solid line) and experimental data (points) from Stern and Gerlach; c) atomic scattering pattern in a magnetic field; d) the same without a magnetic field. Comparing the scattering figures in Figs. 1 and 2, we find that Bohr’s hypothesis about quantizing the magnetic moment with projection $pmmu _B$ to the direction of the magnetic field is confirmed.

A similar result can be obtained without this hypothesis, simply using the Schrödinger–Pauli equation:
$$ihbar frac partial psipartial t=-frac hbar ^22mnabla^2psi+mu _B(vec B.vec sigma )psi $$
$vec sigma =(sigma _1,sigma _2, sigma _3)$ there is a Pauli matrix. Figure 3 shows the scatter pattern in the quantum model.

There are semiclassical equations of motion that fully explain the Stern-Gerlach experiment, see e. g. Section 5 of Gat, Lein & Teufel, Annales Henri Poincaré 15, 1967 (2014). It is important, though, that in this approach it is not the individual trajectories that matter, but averages of an ensemble of initial spins (which are obtained from a generalized Wigner transform of the quantum spin state). And this is how quantum mechanics comes into play, because only states from quantum mechanics give you a consistent interpretation.