I'm trying to get an idea of what spin in quantum mechanics means. I have the following questions regarding spin:
The eigenvalues of the operator corresponding to the z component of spin is $\hat{S}_{z}$ with eigenvalues $\frac{\hbar}{2}$ and $-\frac{\hbar}{2}$ respectively. We then have two corresponding eigenvectors $|\frac{1}{2}(\frac{1}{2}) \rangle$ and $|\frac{1}{2}(-\frac{1}{2}) \rangle$. As I understand from the postulates of QM, any quantum state can be expressed as a linear combination of these eigenvectors. What I don't understand is how we know that these eigenvectors can be expressed as $\chi_+ = \left( \begin{array}{c} 1\\ 0\\ \end{array} \right) \text{ and } \chi_- = \left( \begin{array}{c} 0\\ 1\\ \end{array} \right)$?
What exactly does it mean for a state to be in z spin up state $\chi_+ = \left( \begin{array}{c} 1\\ 0\\ \end{array} \right)$ where it has eigenvalue $\frac{\hbar}{2}$? Is the following the correct interpretation: Does it simply mean that the spin around the z axis can take only two values $\frac{\hbar}{2}$ and $-\frac{\hbar}{2}$ which corresponds to counter clockwise spin around the $\hat{z}$ axis (spin up) so that the vector points up the z axis and respectively counter clockwise spin around the $- \hat{z}$ axis (spin down) so the vector points down the z axis?
If this is the correct interpretation then how does the form of the eigenvectors $\left( \begin{array}{c} 1\\ 0\\ \end{array} \right) \text{ and } \left( \begin{array}{c} 0\\ 1\\ \end{array} \right)$ correspond to a vector up the z axis and one down the z axis?
Thanks for any assistance.