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My professor during the lecture said exactly the following

Let there be a system of non-interacting fermions. Since they are indistinguishable, they have the same Hamiltonian, and the single-particle energies are $E_1, E_2, E_3,\dots$, corresponding to states $S_1, S_2, S_3$, and so on. Since fermions obey Pauli's exclusion principle, the number of particles with energy $E_i$ will be at most 1".

I didn't ask him the following question (thinking it might be rude) what if there is degeneracy? that is what if two different particles in two different states $S_i$ and $S_j$ have the same energy $E_j$?

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Two identical fermions cannot be in the same state. However, if there is a degeneracy, then two identical fermions can have the same energy, so long as they are in different states. This is why, in chemistry, the Pauli exclusion principle allows you to place two electrons in a given atomic energy level. Since one electron has spin up, and the other spin down, the electrons are in different states, even though these states have the same energy (at least approximately).

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