The example I'm trying to understand is:

$ \hat{S}_{x} \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
\end{pmatrix} = 1/2 \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
\end{pmatrix} $

My interpretation of this is that the vector shows you the probabilities of a particle being spin up or spin down if you square them.

And I've been told that $ \hat{S}_{x} $ gives you the spin as an eigenvalue, but how? Since its 50:50 of getting -1/2 and 1/2. $ \hat{S}_{x} $ has only given you one of them.

Is it that $ \hat{S}_{x} $ only measures the magnitude of spin in the x direction?

This post imported from StackExchange Physics at 2014-03-22 16:56 (UCT), posted by SE-user 9k9