The Cartesian components of the spin operators $S_x, S_y$ and $S_z$ don't commute $[S_i,S_j] \neq 0 \ (i \neq j)$.
Hence we can't simultaneously determine all Cartesian components of the spin angular momentum of a spin particle, since the operators of the respective observables at hand don't have a common eigenbasis.
Taking this into account, what do we mean by $\langle \mathbf S \rangle$ ?
Furthermore, if we consider a particle of spin $1/2$ at rest in a uniform magnetic field in the $z$-direction $ \mathbf B = B\ \hat{z}$, where the time evolution of the particle is represented by the spinor $\chi (t) = \begin{pmatrix} ae^{i\gamma Bt/2} \\ be^{-i\gamma Bt/2} \end{pmatrix}$,
what do we mean quantum-mechanically by the observation that $\langle \mathbf S \rangle$ precesses about $\mathbf B$ in the $xy-$plane?
This seems like a classical observation, where we can indeed just determine all Cartesian components at once. But what do we mean by this in the context of quantum mechanics?