In filling the k-space of a metal with electrons, each grid point in the k-space can be occupied with two electrons, one in a spin up state and one in a spin down state. Energy of the electron does not depend on the spin state of the electron. The full quantum state of the electron then includes the position wavefunction and the spin orientation. This can be notates as follows:
The spin of a particle can also be represented as a two-element matrix or spinor, with spin up represented as (1;0) and spin down as (0;1).