Usually in the problems I have done, I found the partition function by simply summing exponential of the eigenenergies, but lately I have started to wonder why this approach is correct. Do we not want to sum over the energies of all possible states as we did in the classical case. In that case we need to take into account that there exists an infinite amount of superpositions of eigenstates, which are all valid states. Are these accounted for when I do calculate the partition function using only eigenenergies? What got me wondering was that my teacher explained the fermi gas by what I thought was a very handwaving argument. He said that its heat capacity is so low because very few electrons can transition to higher energy states because usually the above states are all occupied. Well now, given the fact that there are an infinite amount of states a particle can be in, does this really provide a good explanation?