# Eigenvalues of spin operator [duplicate]

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?

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: $$\pm \hbar/2$$.