Eigenvalues of spin operator

Question

I am currently referring Sakurai. He introduces spin states and operators from general arguments and experimental evidence but ad hoc introduces that the eigenvalues of the pure states as $\pm \frac{\hbar}{2}$, Is there a more foundational argument to why the eigenvalues take those values or can they be scaled arbitrarily?

Answer 1

Given the action of $S_\pm$, which can be constructed from $S_x$ and $S_y$, the eigenvalues of $S_z$ must differ by one $\hbar$ and there are two of them as spin matrices are $2\times 2$ matrices. Hence the eigenvalues are $\pm\hbar/2$. That’s a purely (Lie) algebraic argument. The key point is that you only need to lower once from the top eigenstate to get the bottom eigenstate, so except for rescaling $\hbar$ you don’t have a lot of freedom here.

If you want quantization about an arbitrary axis, then $S_z\mapsto US_zU^{-1}$ where $U$ is a unitary $2\times 2$ matrix. As conjugation by $U$ does not change eigenvalues, it is immediate that quantization about any axis (including $\hat x$ or $\hat y$) will result in only two eigenvalues: $\pm \hbar/2$.

See also this question.