Partition function of two spin 1/2 particles - Distinguishable or indistinguishable? Suppose I have some fermions with spin 1/2 on a harmonic potential. Then the energy of each particle is given by:
$$
E_i=\hbar\omega(n_{x_i}+n_{y_i}+n_{z_i}+3/2)
$$
By definition the partition function is:
$$
Z=\sum_{\{\vec\mu\}}\exp(-\beta E(\vec\mu))
$$
where $\vec\mu$ is the microstate of the system. Being fermions I know that both can't possibly be in the same quantum state at once. Then the sums would be something like $\sum_{n_{x_i}\neq n_{x_j}}$, but my major doubt comes on the accounts of distinguishability, because those particles are only distinguishable if their spin quantum number, $m_s$, is the same for both particles, but if that isn't the case then the particles are in fact distinguishable and than I am not even sure that the fact that they can't be in the same quantum state would be significant because having different quantum spin numbers, it is impossible for them to be in the same quantum state, as they belong to different parts of the joint Hilbert space. How does one account for this?
 A: Particles are not distinguishable or indistinguishable based on their state but rather whether they are excitation of the same field. When we not want to involve this level of abstraction, we simply state (or given) if particles are distinguishable or not. Two particles of the same type will be indistinguishable, in general.
So the question of whether your two fermions are distinguishable or not is not related to the projection of their spin or to the energy level they occupy. These are details of the state. Rather, if the particles are identical (that is, both are electrons, or neutrons, or muons etc.) then we will treat them as indistinguishable.
Tow indistinguishable fermions really cannot occupy the same state. This is because we require that their full wave-function will be anti-symmetric under their exchange. Here the state they occupy come into play. If both of them have the same spin projection (identical $m_z$) then their spatial wave function must be antisymmetric. If, on the other hand, their spin projection is anti-symmetric, then their spatial wave function must be symmetric! (so the total wave function will be anti-symmetric under exchange).
