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Suppose we have an electron in a state given by $|\Psi\rangle = |+\rangle$, so that we know, with 100% certainty that $\hat S_z|\Psi\rangle = \frac \hbar 2 |\Psi\rangle$. But, if we were to measure, say $\hat S_x$ and then $\hat S_z$ again, we could possibly get a different answer (think of the Stern-Gerlach experiment). So, can we then really say that the electron's spin is pointing in the $z$-direction, or must we content ourselves with saying only that the $z$-component of the electron's spin is pointing up?

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  • $\begingroup$ Read your QM intro text (Feynman vIII ?). If you measure with $\hat S_x$ first, and note the eigenvalues and which state you got, your state is not an $\hat S_z$ eigenstate anymore, and it is equally probable to be the up or down state in the z-direction. $\endgroup$ Jan 30, 2022 at 21:07
  • $\begingroup$ @CosmasZachos Yes, I know this, that's why I mentioned it in the question. My question was whether we can consider the spin as truly pointing in the $z$-direction, or whether we can only speak of the $z$-component of the spin as being up. $\endgroup$ Jan 30, 2022 at 21:15
  • $\begingroup$ Linked. $\endgroup$ Jan 31, 2022 at 8:44
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    $\begingroup$ Remember that the total spin is $\hbar \sqrt{3/4}$ so even an electron in the $S_z=+1/2$ eigenstate is not 'pointing in the z direction' but at quite a large angle to it $\endgroup$ Jan 31, 2022 at 21:54

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I think baked into your question is the idea that there is a "spin vector" $\vec{S}$, and you want to know whether the Stern-Gerlach apparatus pointed in the $z$ direction measures $\vec{S}$ or only $\vec{S}\cdot \hat{e}_z$.

While classically this is a perfectly reasonable question, we have to remember that in quantum mechanics we need to be careful that some quantities are not observable even in principle. Because of the angular momentum commutation relations, it is only possible to measure one of $S_x, S_y,$ or $S_z$. Therefore, it is not possible to measure the spin vector $\vec{S}$ precisely, and so I would argue that we should not even really treat it as a physical quantity (though it can be useful mathematically).

There are some other, perhaps more intuitive facts that align with this basic point. First, whenever you measure a component of the spin of a spin-1/2 particle, you will never find it is zero (since you always find it takes one of the values $\pm \hbar/2$). So we can't say that there is no component of spin along a given axis. Second, if the total angular momentum quantum number is $j$, then the magnitude of the spin is $\hbar \sqrt{j(j+1)}$ (this is the square root of the eigenvalue of the $S^2$ operator). This is strictly greater than the largest eigenvalue of the $S_z$ operator, which is $\hbar j$. So whenever you measure the spin component along the $z$ axis, you always get an answer which is less than the total spin magnitude. Classically, this would imply that there must be angular momentum along either the $x$ or $y$ axis as well (or both). Quantum mechanically, we can't quite say that, since we can't ever measure more than one component simultaneously, but (provided you keep the caveats in mind) I think it's useful intuition to think that in a Stern-Gerlach setup, we can only ever measure part of the total spin, and there are other, uncertain spin components we can't directly access.

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  • $\begingroup$ Thank you for your answer. So just to clarify, it would be a misnomer to say that an electron's spin is up, and we should rather say that the component of its spin in the $x$, $y$, or $z$ direction is up? $\endgroup$ Feb 3, 2022 at 16:53
  • $\begingroup$ @DanielWaters That's right; "spin up" and "spin down" always refer to the component of spin along a given axis. $\endgroup$
    – Andrew
    Feb 6, 2022 at 21:47
  • $\begingroup$ Great, thanks for clarifying! $\endgroup$ Feb 6, 2022 at 22:51
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I can't tell the difference in your ontological alternatives,

whether we can consider the spin as truly pointing in the 𝑧-direction, or whether we can only speak of the 𝑧-component of the spin as being up.

In physics, the two are synonyms.

Your spinor (1,0) may be thought to point to the north pole, if that helps you (but why should it?).

If you started with a normalized version of (1,1), you'd see that a $\pi/2$ rotation of your detector around y would align the detector with x, and identify your state with the up x-axis eigenstate, so you might consider your spinor to be pointing to the +x direction. But why?

The Bloch sphere of your state tells you how to identify the "direction" of the spinor for any spinor.

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