Why are the eigenvalues of $S_x$, $S_y$, $S_z$ operators in Stern-Gerlach experiment for spin the same?

Question

I get that $S_z$ operator has spin $\frac{h}{2\pi}$ ,$\frac{-h}{2\pi}$ because the spin angular momentum along the $z$ direction is the latter. But why should this be the same even for $S_x$. Can you explain the intuition?

Answer 1

Invariance of the physical systems under rotation allows the operators $S_z$ and $S_x$ be rotationally equivalent means they can be used one another under a rotation operation. The roots of the rotational invariance can be shown by Noether's theorem. On the other hand if you write the spin operators in matrix form and using a rotation matrix you will see that this transforms $S_z$ to $S_x$.

Answer 2

Qualitatively, the reason is simple. You distinguish between $S_z$ and $S_x$ because of axis orientation, but in case you rotate your axis, $S_z$ becomes your $S_x$ and vice-versa. Thus these two are equivalent, and the eigenvalues must be the same.