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• okaythanksbud
okaythanksbud
I just learned about the Stern-Gerlach experiment and have some questions:

1: clearly there's no objective "up" or "down"--the directions are measured relative to the magnetic field, correct? And well always find just 2 spots of equal and opposite distance on the detector, implying the magnetic moment of the electron always has z component (well take z as the direction the B field is pointing) of either +/-K for some value K?

2: if our B field only varies in the direction it points (i.e if it points in the z direction it'll change only over the z direction), our results will only tell us about the spin in the z direction. is there anything we can infer about the spin in the other 2 dimensions? I'm assuming it is still in superposition since we dont actually measure these components, but in general can the spin in either of the two dimensions be nonzero? My question is essentially whether or not upon measurement the particle aligns its spin entirely with the B field. This would make some shred of sense since its potential energy would be extremized but it seems too idealized

3: is the presence of only two dots true in general? or can the spin take on different values like the energy of a particle in a box? Im assuming we'll get a discrete spectrum but am wondering how many values it can take.

4: if (2) is not true (dipole moment can have nonzero components orthogonal to the B field), wouldn't induced torque complicate the experiment? Im also assuming this would cause a range of values on the detector since the torque would change the z component of the magnetic moment over time, so discreteness doesn't seem like it would hold (there would have to be a cluster around.

5: again, if (2) is not true, can't we determine the total spin using a magnetic field like B=<c1x,c2y,c3z>? This should measure the spin in all 3 dimensions. would the total angular momentum be quantized like <Lx,Ly,Lz> with each L only being able to take on a discrete set of quantities?

So overall I'm wondering if angular momentum in a particle is just quantized in each dimension (so that if measured the spin can take on a wide range of values, but discrete), and if the SG experiment just uncovers the spin in one dimension to make the interpretation simpler.

okaythanksbud said:
the directions are measured relative to the magnetic field, correct?
Yes.

okaythanksbud said:
well always find just 2 spots of equal and opposite distance on the detector
In an ideal experiment, yes. As you will see if you look up online discussions, the results of the actual experiment were not ideal.

okaythanksbud said:
is there anything we can infer about the spin in the other 2 dimensions?
No.

okaythanksbud said:
is the presence of only two dots true in general? or can the spin take on different values like the energy of a particle in a box?
For spin ##j##, the number of dots in an ideal experiment will be ##2j + 1##. In more technical language, that is the number of eigenvalues of the operator for spin ##j## in a particular direction. (And to be even more technically correct, this is only true for massive particles. For massless particles, like photon, there are only two spin eigenvalues--usually called "polarizations"--regardless of ##j##.)

vanhees71
PeterDonis said:
For spin ##j##, the number of dots in an ideal experiment will be ##2j + 1##. In more technical language, that is the number of eigenvalues of the operator for spin ##j## in a particular direction. (And to be even more technically correct, this is only true for massive particles. For massless particles, like photon, there are only two spin eigenvalues--usually called "polarizations"--regardless of ##j##.)
My professor said that only two dots are observed, is there a reason for this? Why wouldn't the be more if the spin can take on more values?

Also regarding my other question, can we measure the total spin, and does this spin have the same quantization in each dimension (i.e each dimension has the same eigenvalues upon having its total angular momentum measured as it does in the SG experiment?)

okaythanksbud said:
My professor said that only two dots are observed,
Observed for what case?

okaythanksbud said:
can we measure the total spin
What does "total spin" mean?

PeterDonis said:
Observed for what case?What does "total spin" mean?
total spin as in the angular momentum vector. The SG experiment should only determine L•z (= L_z) since the magnetic moment in the direction of the B field should be the only thing that contributes to the force. Im wondering if we can also measure L_y and L_x and if they'd be quantized the same way as L_z.

I just looked it up and apparently the spin operators do not commute so I guess we can't measure all 3 simultaneously. I'm assuming S_x,S_y,S_z all have the same eigenvalues though so im assuming the answer would be that they are all quantized the same way.

We didn't cover any "case", my professor just described an apparatus which sent electrons through a specially varying magnetic field which resulted in only two dots being observed. Id assume that regardless of the setup wed observe more than two dots since the spin has multiple eigenvalues so there should be a probability of observing each

okaythanksbud said:
total spin as in the angular momentum vector.
The spin angular momentum vector, I assume (there is also orbital angular momentum and total angular momentum). There is an operator representing the (squared) magnitude of that vector, which is usually written ##S^2 = S_x^2 + S_y^2 + S_z ^2##.

However, as you note:

okaythanksbud said:
I just looked it up and apparently the spin operators do not commute so I guess we can't measure all 3 simultaneously.
That's correct, and it means that the above ##S^2## operator has to be treated very carefully as far as its physical interpretation. You will find plenty of discussion of this in QM textbooks.

okaythanksbud said:
I'm assuming S_x,S_y,S_z all have the same eigenvalues though
Yes.

okaythanksbud said:
We didn't cover any "case"
Yes, you did:

okaythanksbud said:
my professor just described an apparatus which sent electrons through a specially varying magnetic field which resulted in only two dots being observed.
Electrons are spin-1/2 particles, so the spin-1/2 case is the case your professor was describing. You certainly cannot conclude from an experiment on electrons that a similar experiment on particles with spins other than spin-1/2 would only have two dots, and I strongly doubt that your professor was making any such claim.

okaythanksbud said:
Id assume that regardless of the setup wed observe more than two dots since the spin has multiple eigenvalues so there should be a probability of observing each
Spin 1/2 has exactly two eigenvalues. The general formula for spin ##j## is the one I gave in a previous post.

PeterDonis said:
The spin angular momentum vector, I assume (there is also orbital angular momentum and total angular momentum). There is an operator representing the (squared) magnitude of that vector, which is usually written ##S^2 = S_x^2 + S_y^2 + S_z ^2##.

However, as you note:That's correct, and it means that the above ##S^2## operator has to be treated very carefully as far as its physical interpretation. You will find plenty of discussion of this in QM textbooks.Yes.Yes, you did:Electrons are spin-1/2 particles, so the spin-1/2 case is the case your professor was describing. You certainly cannot conclude from an experiment on electrons that a similar experiment on particles with spins other than spin-1/2 would only have two dots, and I strongly doubt that your professor was making any such claim.
I mean that he didn't say anything about there being different cases. If by case you mean type of particle then yes, an electron was used. I was confusing things I was poorly taught in chem classes with your previous statement that particles with spin j has 2j+1 eigenvalues for spin but understand now. It seems kind of strange that it can only take on 2 values, the only eigenvalues spectrums I've seen so far have been infinite, even if discrete. Is there any reasoning/explanation as to why certain particles have more eigenvalues than others or is this just some pattern we happened to notice?

okaythanksbud said:
I mean that he didn't say anything about there being different cases.
So what? He specified what he was talking about: electrons. That means you can't conclude anything about particles of other spin from what he said. If you want to know his answer to the question of how many dots there would be for particles of other spin, you need to ask him that specific question.

okaythanksbud said:
Is there any reasoning/explanation as to why certain particles have more eigenvalues than others
Yes, but it involves group theory, specifically the group representation theory of SU(2). See, for example, here:

https://en.wikipedia.org/wiki/Repre...ucible_representations_and_their_applications

The ##2j + 1## rule is actually stated (in different words) in the introduction to that Wikipedia article, but the section I linked to describes the actual representations that are used for spin-1/2 and spin-1 particles.

vanhees71
okaythanksbud said:
We didn't cover any "case", my professor just described an apparatus which sent electrons through a specially varying magnetic field which resulted in only two dots being observed.

okaythanksbud said:
If by case you mean type of particle then yes, an electron was used.

Just a note: The standard Stern-Gerlach experiment is done with neutral particles (for example, silver atoms), not electrons. You could send electrons through the magnet, but they would be deflected by the magnetic field, making the experiment a bit messy.

vanhees71 and PeterDonis
okaythanksbud said:
My professor said that only two dots are observed, is there a reason for this? Why wouldn't the be more if the spin can take on more values?

Also regarding my other question, can we measure the total spin, and does this spin have the same quantization in each dimension (i.e each dimension has the same eigenvalues upon having its total angular momentum measured as it does in the SG experiment?)
The spin of a particle is fixed. For an electron it's ##s=1/2##. This implies that any spin component can take only two possible values, ##\pm \hbar/2##.

I don't know, what you mean by "total spin". You can simultaneously determine ##\vec{s}^2## (which has the fixed value ##s(s+1)\hbar^2##, and ##s## can take the values ##0##, ##1/2##, ##1##, etc.) and the spin component in one direction, for which one usually chooses ##s_3##. It can take the values ##\sigma \hbar## with ##\sigma## taking the ##-s##, ##-s+1##,...,##s-1##,##s##.

okaythanksbud said:
my professor just described an apparatus which sent electrons through a specially varying magnetic field which resulted in only two dots being observed
The S-G experiment was performed using atoms, not electrons. Because of their charge and low mass, electrons would be deflected by the evXB force, which would convert the two dots to two overlapping circles, making the experiment exceedingly difficult. I think it may have been done with electrons recently in a 'tour d' force'.

Last edited:
vanhees71
I'm not so sure that the historical SG experiment can be done with electrons, but you can interpret also the motion of an electron in a Pennig trap as a kind of SG experiment. This one is among the most accurate measurements ever done.

## What is the Stern-Gerlach experiment?

The Stern-Gerlach experiment is a fundamental experiment in quantum mechanics conducted by Otto Stern and Walther Gerlach in 1922. It demonstrated that particles such as electrons have quantized angular momentum (spin). In the experiment, a beam of silver atoms was passed through a non-uniform magnetic field, causing the beam to split into discrete parts, which provided direct evidence of quantized spin states.

## What was the significance of the Stern-Gerlach experiment?

The Stern-Gerlach experiment was significant because it provided empirical evidence for the quantization of angular momentum, a key concept in quantum mechanics. It showed that particles possess intrinsic angular momentum (spin) that can take on only certain discrete values, laying the groundwork for the development of quantum theory and the understanding of electron spin.

## How does the Stern-Gerlach apparatus work?

The Stern-Gerlach apparatus consists of a source of particles (such as silver atoms), a non-uniform magnetic field, and a detector screen. As the particles pass through the magnetic field, their magnetic moments interact with the gradient of the field, causing the particles to deflect in different directions based on their spin states. The detector screen then captures the spatial distribution of the deflected particles, revealing discrete spots corresponding to different spin orientations.

## What did the results of the Stern-Gerlach experiment show about the nature of spin?

The results of the Stern-Gerlach experiment showed that spin is quantized, meaning that particles can only have specific, discrete values of angular momentum. For silver atoms, the experiment revealed two distinct spots on the detector screen, corresponding to the two possible spin states (spin-up and spin-down) of the electrons. This provided clear evidence that spin is an intrinsic property of particles and is not continuous.

## Why was silver used in the original Stern-Gerlach experiment?

Silver was used in the original Stern-Gerlach experiment because silver atoms have a single unpaired electron in their outer shell, making them suitable for studying the effects of spin. The magnetic moment of the unpaired electron interacts strongly with the non-uniform magnetic field, leading to a clear and measurable deflection. Additionally, silver atoms are relatively easy to vaporize and manipulate in an experimental setup.

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