Eigenstates of Sz
Since the z component of a spin-1/2 particle can take on just two values, +ħ/2 or −ħ/2 (quantum numbers ms = +1/2 or −1/2), we often call these states "spin up" or "spin down" but we are actually referring just to the z component of spin. Since there are two values, we can write these states as two-component column vectors which are eigenstates of Sz:
These states are clearly eigenstates of Sz which can be verified by direct calculation. These states can be explicitly determined by using linear algebra to determine the eigenvalues and eigenvectors of the Sz matrix.
Eigenstates of Sx and Sy
It is also of interest to write down the eigenstates of the other spin operators. We can write such eigenstates in terms of eigenstates of Sz if we like. This is called a choice of basis or choice of basis vectors. In particular, we find that the eigenstates of Sx are:
or that
We also have that
and finally
The form of these states should convince you that eigenstates of one component of spin, will not be eigenstates of the other components of spin.