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$\newcommand{\ket}[1]{|#1\rangle}$ Since the state any spin-1/2 particle can be written as a linear combination of $\ket{+z}$ and $\ket{-z}$ eigenstates of the $z-$direction spin operator $S_z$, we can write any $\ket{+\hat{n}}$ state in terms of $\ket{\pm z}$, with $\ket{+\hat{n}} = \cos(\theta/2)\ket{+z} + e^{i\phi}\sin(\theta/2)\ket{-z}$ up to a phase. I just wanted to confirm that:

  1. For any spin-1/2 particle, there exists some direction $\hat{n}$ in $\mathbb{R}^3$ such that measuring the spin in that direction will give a definite $+\hbar/2$ measurement, i.e., the particle's state is purely $\ket{+\hat{n}}$.
  2. Is it correct to think about a 2-dimensional spinor $\ket{+\hat{n}}$ containing enough information about an orientation $\hat{n}$ in 3 dimensions (identified by ($\theta$, $\phi$))? Is this property unique to spin-1/2 systems?
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Yes, you can always think of a spin 1/2 particle as "pointing" in some direction of space, which is used in common visualizations like the Bloch sphere. But this doesn't work for higher spins, because the set of directions has dimension $2$ and the spin states have dimension $2s+1$.

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