When was the formula for the Stern-Gerlach experiment found?

[Frigorifico9]

TL;DR Summary

The formula for the splitting of the beam is: cos^2(theta/2), but I can't find any mention of this formula in paper from that time

Today we know that if you make successive Stern-Gerlach measurements the beam of atoms will split according to this formula:

> cos^2 (theta/2)

And this is something people back then could have figured out, they could have done many measurements, plotted the values, and realized it followed this particular function, even if they didn't understand yet why it happened

But here's the thing, I've been going very in depth on this topic (sources linked below) and I can't see any reference of people discovering this formula

This is odd because people like Rabi were inspired by this to make other experiments that were much more complex, so they had the means to make the measurements and discover this formula, but they just don't mention it

Can any of you hook me with some sources that mention this formula? Or was this formula not discovered until after Pauli had found his matrices?

Some sources I have read that don't mention this formula anywhere:

Pauli's original paper where he introduces the Pauli matrices

This paper explaining the Stern-Gerlach experiment with our modern understanding

The original paper by Stern and Gerlach

The history of the Stern-Gerlach experiment

 

[PeterDonis]

successive Stern-Gerlach measurements

The original Stern-Gerlach experiment didn't make successive measurements. It didn't take one of the output beams of an S-G device and put it through another S-G device with a different orientation. So of course that experiment (and many other experiments like it which didn't do what I just described) would not tell you anything about the formula you refer to. As far as I can tell, none of your references discuss successive S-G measurements, so of course they won't say anything about that formula either.

 

[Frigorifico9]

Thank you, but please don't stop at pointing out the flaws in my reasoning, could you help me actually find a source for the origin of this formula? Because despite how flawed and stupid my reasoning was, that is my goal

 

[PeroK]

Thank you, but please don't stop at pointing out the flaws in my reasoning, could you help me actually find a source for the origin of this formula? Because despite how flawed and stupid my reasoning was, that is my goal

That formula may be more relevant to the tests of Bell's Theorem than SG.

 

[PeterDonis]

could you help me actually find a source for the origin of this formula?

Now that I have pointed out the flaw in your reasoning, have you tried modifying your "in depth" search accordingly?

 

[Frigorifico9]

Now that I have pointed out the flaw in your reasoning, have you tried modifying your "in depth" search accordingly?

Obviously, otherwise I wouldn't need the help of people smarter than me, like you. I've been checking papers by Rabi, Frish, and Segre. They performed an experiment using three fields, and the middle one was rotating, but nowhere in the papers do they mention this formula

Here's the DOI of the article in question: 10.1038/130892d0

 

[Vanadium 50]

"Tell me where Newton wrote E = mc². Don't tell me where I an wrong, just answer my question!"

The formula you posted, cos²(θ/2), has four maxima. The simple Stern-Gerlach experiment has two. So those two cannot go together.

 

[vanhees71]

The SG experiment in its original form is quite difficult. It's amazing, what Stern and Gerlach did in 1921! At this time they didn't do successive Stern-Gerlach experiment nor did they know the correct theory, i.e., modern quantum theory and the notion of spin, let alone spin 1/2. What was, however, done, as one can see from the original papers by Stern and Gerlach, is a measurement of the magnetic moment of the electron, based on the classical formula for deflection of a magnetic moment in an inhomogeneous magnetic field. In hintsight this works, because to a good approximation it's just the motion of a particle in a constant force field, i.e., according to Ehrenfest's theorem the equation of motion for the expectation value of the Ag atom's position is exactly the classical equation of motion, $m \ddot{\vec{x}}=\vec{F}=\text{const}$.

What Stern and Gerlach figured out was that the deflection was compatible with a magnetic moment of 1 Bohr magneton. This was to some extent expected by using the "old quantum theory" (Bohr-Sommerfeld quantization), for the analogue of what we'd call spin 1 today. Of course, what was strange is that they found only 2 lines and not 3, but this was also somehow handwavingly explained away by Bohr. That 1 Bohr magneton in fact is correct is because (a) electrons (and thus the AG atom) have spin 1/2 and (b) the gyro-factor of the electron is very close to 2. Of course, it took quite a while until this was figured out in connection with the Zeeman effect(s).

For the amusing story about how Stern and Gerlach achieved all this in 1921/22, see

https://faculty.chemistry.harvard.edu/files/dudley-herschbach/files/how_a_bad_cigar_0.pdf