I've started reading many-particle systems in Quantum mechanics, and came across the concept of identical particles vs distinguishable particles.
However, I'm wondering, what happens in case of a set of distinguishable fermions. Are their any special rules, like the pauli exclusion principle for identical fermions, that we need to keep in mind while filling up these particles ?
Say, we have 5 distinguishable fermions of same mass in a $1D$ harmonic oscillator. Since the particles are distinguishable, we can use separation of variables to separate the wavefunctions for the $5$ fermions.
Suppose the system is in its ground state. Hence, the $5$ distinguishable fermions must also be in their respective ground states. However, since all of them have the same mass, they would have the same ground state energy level. Thus, in the system, we have an energy level that has $5$ distinguishable fermions in the ground state.
Aren't the fermions in here, just behaving as identical bosons would do in the same potential ? Is my intuition correct, or does the fermions not fill up their respective ground states in this manner. If they were identical, this would not have been the case, as it would have violated the exclusion principle. However, does the exclusion principle come to play even in case of distinguishable fermions.
From the energy level perspective, Distinguisable fermions seem to act exactly the same way as any distinguishable particles. The only difference comes in the wave functions, as we now have to consider the spin states too. However, from the energy level perspective, am I correct in saying that identical bosons, distinguishable particles and distinguishable fermions of the same mass, have the exact same energy values for different states and only their wave functions are different ?