Question about Stern-Gerlach experiment

[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.

 

[PeterDonis]

the directions are measured relative to the magnetic field, correct?

Yes.

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.

is there anything we can infer about the spin in the other 2 dimensions?

No.

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 $j$.)

 

[okaythanksbud]

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?)

 

[PeterDonis]

My professor said that only two dots are observed,

Observed for what case?

can we measure the total spin

What does "total spin" mean?

 

[okaythanksbud]

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

 

[PeterDonis]

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:

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.

I'm assuming S_x,S_y,S_z all have the same eigenvalues though

Yes.

We didn't cover any "case"

Yes, you did:

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.

 

[PeterDonis]

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.

 

[okaythanksbud]

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?

 

[PeterDonis]

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.

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.

 

[Doc Al]

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.

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]

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 $-s$, $-s+1$,...,$s-1$,$s$.

 

[Meir Achuz]

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'.

 

[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.