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As I understand it, the Pauli Exclusion principle states that two electrons in orbitals of a given atom with the same values for quantum numbers $n$, $l$ and $m_j$ must have different (opposite) values for $m_s$, their spin. My question is: what is the physical interpretation of having "opposite spin", considering that spin can be measured along any axis? Does this means that if one measured the spins of the two electrons in any direction, they would always be antiparallel - or does it mean that there is some inherent axis along which the atom "measures" the spin of the two electrons and only allows them into the same orbital if they have opposite spins?

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It's your first interpretation. The spin-singlet is the unique state of $s=0$, $m_s=0$, which is an eigenstate of $S_z$, $S_x$, and $S_y$ simultaneously, and in fact $$ \lvert 00\rangle = \frac{\lvert \uparrow \downarrow \rangle -\lvert\downarrow \uparrow \rangle}{\sqrt{2}} = \frac{\lvert \leftarrow ,\to \rangle -\lvert\to,\leftarrow \rangle}{\sqrt{2}} = \frac{\lvert s_{\hat{n}}=\uparrow,s_{\hat{n}}=\downarrow \rangle -\lvert s_{\hat{n}}=\downarrow,\lvert s_{\hat{n}}=\uparrow \rangle}{\sqrt{2}} \,, $$ where $\hat{n}$ is any direction; which is to say, it is the antisymmetric combination of up-and-down states along any axis. Thus, yes, if you measure the spins of the two electrons along any axis, they will always end up being antiparallel.

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    $\begingroup$ This is correct, of course, but it is worth pointing out the real reason it is correct: a more general formulation of the Pauli exclusion principle is that the two-electron wavefunction must be antisymmetrized. If the spatial part for each electron is the same, this is then the only possibility for the spin part. $\endgroup$
    – Rococo
    Oct 19, 2022 at 21:23
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    $\begingroup$ @Rococo I took this as more a question about what the spin singlet is versus why it has to be in a spin singlet to begin with, but your point is well taken. $\endgroup$
    – march
    Oct 19, 2022 at 21:25

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