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This question is similar to this post and was influenced by this discussion as well.

Assume a Stern-Gerlach apparatus oriented along the z-axis acts as a measurement on the basis of the z-component of a spin-1/2 particle's spin. The incoming particles travel along the y-axis and have randomly aligned magnetic moments / spins.

If we focus on one single particle from the incoming beam, say with spin pointed along the x-axis, will its spin be reoriented into the $\pm$ z direction when it passes through the magnet? Is it incorrect to assume the spin state of one of the particles before it passes through a magnet?

I think each incoming particle's spin is supposed to be treated as a superposition of states. I suppose treating an incoming spin as being in some superposition of $\pm$ z states could account for any orientation of the spin, including x- or y-components. Is this the case? And then the measurement causes the particle's hbar/2 units of angular momentum to be assigned to either $\pm$ z?

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To say that the spin "is pointed along the $x$-axis [and will] be reoriented into the $\pm z$-direction" isn't quite right to begin with. A quantum-mechanical spin doesn't exactly point in a specific direction; it's just that it only has a definite projection onto (at most) a single axis.

Indistinguishable particle questions aside, you can say that an incoming particle has a definite spin projection along the $x$-axis, but that's the most you can say. In that case, the projection of its spin onto the $z$-axis is ill-defined; it's in an admixture of states of definite projections onto the $z$-axis.

The interaction with the magnetic field of the detector then puts the particle into one paticular one of those states with definite projections onto the $z$-axis, but in doing so, puts it into a similar admixture of states with definite projections onto the $x$-axis, so that now the projection of the particle's spin onto the $x$-axis is ill-defined.

Yes, you are right, the state with a definite projection onto the $x$-axis (or the $y$-, or whatever) is a superposition of states with different projections onto the $z$-axis. In that case, the superposition breaks down and the projection of the particle's spin is "assigned" to the $z$-direction, but to do so, it must be "unassigned" from the $x$- (or $y$-, or whatever) direction.

As for assuming the spin state of one of the particles, you run into indistinguishable particle questions if you try to consider a single particle in an ensemble, so you really do need to be talking about the full ensemble.

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