I was going through a quantum objects course on brilliant.org, a few things have been bothering me and though I searched A LOT I can't seem to find any answers. For one, the course implies that a neutron going through a magnetic field aligned on the z-axis will come out of the magnet as spin up and spin down and says that this is due to the spin of the neutron. I've looked it up and people seem to be directing the magnetic moment of the neutron not to its spin properties but to the moment of the quarks within it. Which one is it? Secondly, 100 particles whose spins are aligned on the positive z axis are taken and passed through a magnetic field aligned with the x axis. This time, it is observed that they come out with their spin aligned either on the positive or negative x axis. They put the ones that came out with their spins aligned on the positive x axis, put it once more in a z-aligned magnetic field and have us observe that they come out with positive and negative spins along the z axis once more. So then, they do the same z-x-z magnet set-up that they had, but they make it so that all the outputs of the x magnet go directly to the z magnet without being observed. Then, they state that now none of them are directed with spin down, because only measurement of a state will change the state itself. This particular phenomenon is a part of quantum physics as far as I can tell, but given that we have an ideal environment and can do these experiments, will these actually hold out the same way we are told they will? Are they just using these as analogies to explain the topic itself or can I actually put a neutron whose spin has been aligned on the positive z axis, not measure it, put it in to an x aligned magnet and it will come out just as it was? Whereas if I observe it, it will come out as aligned with positive and negative x? Shortly, are the things they are saying experiments that hold true?

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    $\begingroup$ Forget about quarks here. The magnetic moment of the neutron is aligned to its spin, so in a magnetic field spin up neutrons have different energies than spin down ones. The rest sounds like a discussion of standard counterintuitive non-commutativity features of QM, best studied in Feynman's undergraduate text, vIII. $\endgroup$ Feb 16, 2021 at 20:38

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The neutron is a spin 1/2 neutral particle. That means the only magnetic property it can have is a dipole moment, and it must be aligned with the spin.

It can have an electric dipole moment, but that is parity and time-reversal violating (see: Electric Dipole Moment of the neutron).

Since it is neutral, but has a magnetic moment, it must have internal structure: valence quarks, sea quarks, gluons, all with spins and possibly orbital angular momentum.

How they create the spin/magnetic moment is an area of active research (The Proton Spin Crisis), but it completely irrelevant for the Stern-Gerlach (SG) experiment.

The question seems to think the quarks can move around and change and affect the spin/magnetic moment. This is not the case: you can flip a quark spin, but then you have a $\Delta^0$ baryon, not a neutron. All neutrons are identical and indistinguishable, there is no wiggle room.

Regarding the SG experiment: yes, all that can be done. Your description starts with an unpolarized mixed state, which requires density matrices. A simpler formulation with all the essential features begins with a beam of up ($+z$) polarized neutrons in a pure state:

$$ \psi = |\uparrow\rangle $$

In terms of the $x$ basis, this is:

$$ \psi = \frac 1 {\sqrt 2}[|\rightarrow\rangle + |\leftarrow\rangle]$$

When that is passed though an $x$-oriented SG device, there are two spatial separate beams:

$$ \frac 1 {\sqrt 2}|\rightarrow\rangle = \frac 1 2[|\uparrow\rangle + |\downarrow\rangle]$$

$$ \frac 1 {\sqrt 2}|\leftarrow\rangle = \frac 1 2[|\uparrow\rangle - |\downarrow\rangle]$$

If you then pass the $|\rightarrow\rangle$ beam through a $z$-SG device, you split into two beams of $\frac 1 2|\uparrow\rangle$ and $\frac 1 2|\downarrow\rangle$. (Note, the $\frac 1 2$ gets squared, so each beam has $\frac 1 4$ of the original number of neutrons.)

On the other hand, if you recombine the beams and pass them through the $z$-SG device, you do not have which-way path information, and you need to add them coherently:

$$ \psi = \frac 1 {\sqrt 2}|\rightarrow\rangle + \frac 1 {\sqrt 2}|\leftarrow\rangle $$ $$ \psi = [\frac 1 2[|\uparrow\rangle + |\downarrow\rangle]] + [\frac 1 2[|\uparrow\rangle - |\downarrow\rangle]] = |\uparrow\rangle $$

and when that traverses the SG device, a single up polarized beam emerges.


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