Why Does the Stern-Gerlach Experiment Show Discrete Deflection Angles?

[gentsagree]

Since the particles aren’t rotating, but just traveling in a straight line, there should be no angular momentum whatsoever. Since the magnetic moment of magnetic dipoles can be theoretically represented as coming from a loop of electric current, or particle orbiting, thus magnetic moments are related to orbital angular momentum.

Now, the Stern-Gerlach experiment sees a deflection of the beam, where deflection should only be due to orbital angular momentum, therefore we infer the existence of an intrinsic (spin) angular momentum. First question: is this correct?

Then we observe a quantization of spin angular momentum, i.e. only discrete possibilities of deflection angles. But how can we say that this is in contrast with orbital angular momentum (as the latter normally gives a continuum of angles), given that we assume there is no orbital momentum in this experiment?

Thanks!

 

[blue_leaf77]

Since the particles aren’t rotating, but just traveling in a straight line, there should be no angular momentum whatsoever.

In Stern-Gerlach experiment, they used atoms, not particles. So, obviously there could be rotational motion, which is in this case due to the orbital angular momentum.

Now, the Stern-Gerlach experiment sees a deflection of the beam, where deflection should only be due to orbital angular momentum, therefore we infer the existence of an intrinsic (spin) angular momentum. First question: is this correct?

As far as I can remember, the atoms used in SG experiment are silver atoms which have 47 electrons. The last unpaired electron occupies 5s orbital, therefore the total orbital angular momentum is zero - if only spin had not existed in the first place, there would be no splitting out of this beam of silver atoms.

But how can we say that this is in contrast with orbital angular momentum (as the latter normally gives a continuum of angles), given that we assume there is no orbital momentum in this experiment?

No, the orbital angular momentum is also quantized.

 

[gentsagree]

Ah, thank you, very clear. Just one thing: if they didn't know anything about spin back then, did they choose an element with the only unpaired electron with l=0 by chance? If not, and they chose it to make the angular momentum zero, what were they expecting to observe?

 

[blue_leaf77]

SG experiment was actually motivated by a wrong assumption and theory. The hypothetical theory of quantum mechanics of Sommerfeld at the time predicted that a quantum system with orbital angular momentum $L=1$ was quantized into two directions of magnetic moment, this is the first mistake for we know today that the z component of $L=1$ angular momentum is discretized three-folds: -1, 0, 1. The second mistake was that Stern and Gerlach assumed that the total angular momentum of silver atoms was unity (which was why they conducted this experiment to test Sommerfeld's theory). Albeit all these errors, the result showed agreement with the hypothesis to be confirmed, namely the beam of silver atoms split into two under magnetic field, as Sommerfeld predicted. They were unaware of the spin of electron at the time though.

 

[gentsagree]

Very clear again. Thank you.

 

[drvrm]

Now, the Stern-Gerlach experiment sees a deflection of the beam, where deflection should only be due to orbital angular momentum, therefore we infer the existence of an intrinsic (spin) angular momentum. First question: is this correct?

Then we observe a quantization of spin angular momentum, i.e. only discrete possibilities of deflection angles. But how can we say that this is in contrast with orbital angular momentum (as the latter normally gives a continuum of angles), given that we assume there is no orbital momentum in this experiment?

Perhaps the following details will help clarify the results of the Experiment:
The experiment is normally conducted using electrically neutral particles or atoms.
This avoids the large deflection to the orbit of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate.
If the particle is treated as a classical spinning dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole (see torque-induced precession).
If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected.
However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory.
If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous.
Each particle would be deflected by a different amount, producing some density distribution on the detector screen.
Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount.
This was a measurement of the quantum observable now known as spin angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or point spectrum.
Although some discrete quantum phenomena, such as atomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to conduct measurements of deliberately superposed quantum states for the first time in the history of science.
See details<https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment>