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Consider two sets of axes, $xyz$ and $x'y'z'$, and the two spin states

\begin{align} |\psi\rangle &= A(|+_x\rangle + |+_y\rangle + |+_z\rangle)\\ |\psi'\rangle &= A(|+_{x'}\rangle + |+_{y'}\rangle + |+_{z'}\rangle). \end{align}

Where $A$ is a normalization constant. Here $|+_x\rangle$ is the spin-up eigenstate along axis $x$, etc. I'm thinking spin $½$, but I don't think that matters.

Obviously, if $x=x'$, $y=y'$ and $z=z'$, these are the same state. Are they the same state if $x=y'$, $y=z'$ and $z=x'$? What about for arbitrary relative orientations of $xyz$ and $x'y'z'$?

If these are always the same state, does that imply that $\langle S_x\rangle = \langle S_y\rangle = \langle S_z\rangle$ for this state? If not, why not? (They're not equal when I calculate them!)

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There's not a unique $|+_z\rangle$, because you can always choose a difference phase. This doesn't matter for the eigenstate, but once you take a superposition the relative phase makes a real difference. So your definitions of $|\psi\rangle$ and $|\psi'\rangle$ are ambiguous. Indeed, $|\psi\rangle$ could be $|+_z\rangle$ for one choice of phase, or could be $|-_z\rangle$ for a difference choice!

In any case, no state in a spin half system can be invariant under any rotation, since all states have some axis along which they have definite spin up.

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  • $\begingroup$ Okay, that makes sense, thanks. Although I think "ambiguous" might be too kind, maybe ill-defined or undefined would be better. I suppose this generalizes? Given two non-commuting operators whose eigenstates span the same Hilbert space (say $x$ and $p$), you can't superpose their eigenstates because of this phase ambiguity. E.g. $|\psi\rangle> = |x\rangle +|p\rangle$ is not a valid state. $\endgroup$ – Chris G. Jan 28 '15 at 20:27
  • $\begingroup$ You certainly can make a superposition of these states, you just have to make sure you define exactly what vectors you mean first. $\endgroup$ – Holographer Jan 29 '15 at 2:57
  • $\begingroup$ As a mathematical exercise, sure, but what about the physical interpretation? The intent of the person who suggested this state (one of my students) was to construct a state that is equal parts spin up in the $x$, $y$, and $z$ directions. But the phase ambiguity means you can create many different states out of "equal parts spin up" in the three directions by manipulating the relative phases. So it doesn't seem like a useful way to construct states to me. $\endgroup$ – Chris G. Jan 29 '15 at 13:25

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