### Alternative Explanation of the Stern Gerlach Experiment

by John David Best, 2015

The results of this experiment, which was first conducted in 1922, are popularly interpreted as demonstrating the quantization of spatial orientation of intrinsic angular momentum. The results can alternatively be explained more simply, without the need for exotic theory.

A diagram of the experimental apparatus (from Wikipedia, Peng 1 July 2005) is below:

In the experiment, a stream or beam of silver atoms passes through an inhomogenous magnetic field. When the stream exits the field, the atoms are deposited on a glass plate. Two separate bands of silver deposited on the plate, show that the beam or stream was split into two streams by the magnetic field. Instead of a complex quantum mechanical explanation, this phenomenon can be explained by the behavior of magnetic dipoles in an inhomogenous magnetic field.

It is generally agreed that atoms can have magnetic moments. For there to be a moment, there must be two poles separated by a distance. This is a magnetic dipole. Silver atoms, which were employed in the experiment, are magnetic dipoles. Dipoles that are subjected to an inhomogenous magnetic field will experience torque until they are aligned with the field. This is because the two ends or poles of the dipole are pulled in opposite directions by the magnetic field. In the current loop model used to describe the dipole moments of atoms, opposite sides of the current loop are pulled in opposite directions.

When the stream of silver atoms in the experiment is exposed to the magnetic field, the magnetic dipoles of the atoms align so that each end or pole of the dipole is closer to the magnetic pole that attracts it than to the pole that repels it. If a dipole is positioned so that its axis of rotation is equidistant from both magnetic poles of the field, the dipole feels no net linear force pulling it in either direction because each end of the dipole is the same distance from the pole that attracts it. If however, one end of a dipole is even slightly closer to the magnetic pole that attracts it, than the other end is to the opposite magnetic pole that attracts it, there will be a net linear force on the dipole, causing it to move toward the side to which it has the strongest net attraction. This could cause a stream of dipoles such as the atoms in the experiment, to separate into two streams.

The net linear force on a dipole is:

*F*_{net}* = F*_{N}* + F*_{S} (eq 1) where *F*_{N} and *F*_{S} are the net forces acting on the North and South poles of the dipole, and

*F*_{N }*= **F*_{N}_{S}* +** **F*_{N}_{N}* *(eq 2) where *F*_{N}_{S} is the force acting between the North pole of the dipole, and the South pole of the external field, and *F*_{N}_{N}* *is the force acting between the North pole of the dipole, and the North pole of the external field.

Likewise, *F*_{S}* **= **F*_{S}_{N}* +** **F*_{SS}* *(eq 3) where *F*_{S}_{N} is the force acting between the South pole of the dipole, and the North pole of the external field, and *F*_{SS}* *is the force acting between the South pole of the dipole, and the South pole of the external field. If *F*_{N}* *and* F*_{S} are both equal in magnitude, then the net force on the dipole is zero, because they act in opposite directions.

Dipoles align so that each pole of the dipole is closer to the pole of the external field that attracts it than it is to the pole that repels it. if *d** *is the distance separating the two poles of the dipole, then the pole of the dipole that is repelled by an external pole, will be at distance, *d*, farther away from the external pole, than the pole of the dipole that is attracted by it.

If *r*_{N}_{S} is the distance between the North pole of the dipole and the South pole of the external field to which it is attracted, then the South pole of the dipole will be at a distance *r*_{N}* *_{S }*+ d *from the South pole of the external field, and likewise, if *r*_{S}_{N} is the distance between the South pole of the dipole and the North external pole that attracts it, then the North pole of the dipole will be at at distance of *r*_{S}_{N}* **+ d *from the North pole of the external field that repels it.

D_{NN }= *r*_{S}_{N}* **+ d** *(eq 4) * *and

D_{SS }= *r*_{N}_{S}* **+ d** *(eq 5)

The force between a pole of a magnetic dipole, and a magnetic pole of the external magnetic field is the force between two magnetic poles, given by

(eq 6)* *where* r *is the distance between the poles.

This equation correctly applies only to idealized point poles, but it suffices to show that the force between two magnetic poles depends inversely on the square of the distance between them. This means that the force between a pole of the external field and the pole of the dipole that is closest to it, is greater than the force between the external pole and the pole of the dipole that is a greater distance, *d*, away.

When the centerpoint of a dipole is exactly mid way between the two external poles, so that each pole of the dipole is equally distant from the pole of the external field that attracts it, the dipole feels no net linear force, because the forces on each pole of the dipole are equal in magnitude due to symmetry. If however, the midpoint of the dipole is even minutely closer to one of the external magnetic poles than the other, then the dipole will be pulled toward the pole that it is closer to as it travels through the inhomogenous magnetic field.

If *r*_{S}_{N}* **≠ **r*_{N}_{S}* *then *F*_{net}* ** **≠ *0. This means that every silver atom dipole in the stream whose centerpoint does not remain exactly at the centerline of the external field (a line equidistant from both poles) as it passes through the field, will be drawn toward one of the poles of the external field. If the stream atoms is directed along centerline of the external field, then we would expect that roughly equal numbers of the silver atoms would be drawn toward each pole. This would cause the stream to separate into two streams with a gap in between. The width of the gap is determined by the magnitude of the forces on the dipoles, and the length of time that the particle travels through the external field.

No further analysis is necessary to be able to say that an external inhomogenous magnetic field can cause a stream of dipoles to separate into two streams. In fact the results of the experiment were exactly what one would expect to result from this mechanism (noting that the external magnetic field is weaker toward the sides in the apparatus as described). Quoting from Wikipedia: “The experiment was first performed with an electromagnet that allowed the non-uniform magnetic field to be turned on gradually from a null value. When the field was null, the silver atoms were deposited as a single band on the detecting glass slide. When the field was made stronger, the middle of the band began to widen and eventually to split into two, so that the glass-slide image looked like a lip-print, with an opening in the middle, and closure at either end.”

If a similar experiment were to be conducted using a stream of particles having multiple discrete values of *d*, then the expected result is that the particles would separate into as many streams or bands on the slide, as there are discrete values for *d*, given a long enough travel time through the external field. each band on the detector slide would be comprised of particles that all have the same magnitude of *d*, with the bands comprised of particles with smaller values for *d* being closer to the centerline of the external field, and those with larger *d* values being farther away. This is because other factors being equal, a particle with a larger *d* will be more strongly attracted to an external pole than one with a smaller *d*.

We know by *F = ma** *(eq 7) that this attraction to an external pole will cause the particle to accelerate toward the external pole, and that the magnitude of the acceleration depends on the force. The distance that the particle moves toward the external pole is given by *x = ½at*^{2}* *(eq 8). so the distance that the particle moves toward the external pole as a function of the net attractive force, is x = F*t*^{2}*/2m** *(eq 9)*.** *If the time that the particle is in the magnetic field is the same for all the particles, as it would be for a stream of particles, then the distance that each particle displaces toward the pole to which it feels a net attraction, is a linear function of the net attractive force. If we plug *d** *into (eq 6) in place of *r,* this gives us the value of the difference in the magnitude of attraction or repulsion toward an external pole, that is felt by the two poles of the dipole. Using this value in (eq 7) gives us the value of the acceleration of a particle toward the external pole to which it feels a net attraction, *a =** **F*_{net}*/m *(eq 10)*.*

By (eq 6). each discrete value of the dipole separation distance, *d* ,maps to a single unique value of *F*_{net}* *, the net linear force acting on the particle. Particles with a larger magnitude of *d* will be pulled farther away from their initial positions relative to the centerline of the external magnetic field than will particles with a smaller magnitude of *d*. If for some reason, possibly due to the structure or degree of excitation of the particle, the distance *d* is held to a finite number of discrete values, this could cause a stream of dipoles traveling through an inhomogenous magnetic field to segregate into as many streams as there are different values of *d**, *given a sufficiently long time of passage through the magnetic field.

In conclusion, it appears that the Stern – Gerlach results can be explained satisfactorily without the need for exotic quantum theory. The experiment involves microscopic or macroscopic objects and forces; the silver atoms constitute real objects that can be seen in aggregate, and there is some real force that causes the stream of atoms to separate. The relatively simple electromechanical explanation presented here seems more appropriate for such a simple phenomenon than any explanation that can only be properly explained mathematically.

**Vabs = Vo + Vr**