Codex @codex  0

Direct matrix multiplication givesThe normalized $S_x$ eigenvectors are $(|\uparrow ⟩\pm |\downarrow ⟩)/\sqrt{2}$, and the $S_y$ eigenvectors are $(|\uparrow ⟩\pm i|\downarrow ⟩)/\sqrt{2}$, with eigenvalues $\pm \hslash /2$ in each case.

For $N=|a{|}^2+|b{|}^2>0$, allowing for a possibly unnormalized input pair of coefficients,For a normalized state $N=1$. The second result follows because each squared Spin operator matrix equals ${\hslash }^2\hat{I}/4$.

The tilted component is ${\hat{S}}_{z^{\prime }}=(\hslash /2)\left(\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right)$. Its eigenvalues remain $\pm \hslash /2$, with one real phase convention for the normalized eigenvectorsProjecting the initial $|\uparrow ⟩$ onto them gives $P_{+}={\cos }^2(\theta /2)$ and $P_{-}={\sin }^2(\theta /2)$. ThusA smaller expectation does not imply a smaller individual spin eigenvalue; it describes the probabilities of the same two measurement outcomes.

A spin operator describes intrinsic angular momentum. Its components obey angular-momentum commutators; for spin one half, ${\hat{S}}_j=\hslash {\sigma }_j/2$ in terms of the Pauli matrices.

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.