Calculating Canonical State Sum with fermions? my question is regarding to the fact that we say that $n=0,1$ for fermions/electrons but why not $n=0,1,2$ if a spin up and a spin down electron can simultaneously occupy the same state?
Thanks for the replies!
 A: Because we count the occupation number for a state in the full single particle Hilbert space (not for orbital states). The full state of an electron is specified by its orbital state and its spin state. That is, there are two states (one for each spin projection) for each orbital state, since the total single particle Hilbert space is the tensor product of the spin states with the orbital states.
This gets especially important if the spatial and the spin parts of the Hamiltonian do not separate (for example, when there is a space dependent magnetic field, or if you include spin-orbit coupling in your calculations). Then you can no longer say that there is a spin up and a spin down state per orbital state, since the eigenstates of the Hamiltonian are no longer tensor products of spin states and orbital states.
Also, the description with $n = 0,1$ also holds for any spin $m/2$ fermions (where, in the case of Hamiltonian where spin and orbit parts decouple) for which there are $m+1$ spin states. And even the artificial spinless fermions we use in our theoretical toy models. (Artificial in the sense that no such thing exists in nature.)
