A question on Fermi free gas model

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When we are calculating the fermi energy, we say that each energy level is 2 fold degenerate and the fermions stack up into higher and higher energy level due to the Pauli Exclusion Principle. My question is: The Pauli Exclusion principle only says that we can't put different fermions into the same eigenstate, so if we have fermions which are in states of a superposition of different eigenstates with different quantum amplitudes, the fermions do not have to go to higher energy levels because it doesn't contradict the Pauli Exclusion principle anymore?
 

alxm

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No, the exclusion principle is always valid. Say you have two particles which are both in a superposition of two eigenstates. You measure a particle in one state, then the other particle must be in the other state.

Or put it this way: A wave function which is a superposition of several eigenstates for several particles that does not give a zero probability of finding two particles in the same state, simply isn't a valid solution to the Schrödinger equation for fermions.
 
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Or put it this way: A wave function which is a superposition of several eigenstates for several particles that does not give a zero probability of finding two particles in the same state, simply isn't a valid solution to the Schrödinger equation for fermions.
I wonder if you mean that it's technically a valid solution of the differential equation but we have to throw it out because it violates the exclusion principle?

For example, if we take the ground state solution of the hydrogen atom and multiply it by any number greater than 1, then the resulting wave function is still a solution of the differential equation. We throw it out because it's not normalized, but it's still a valid solution of the differential equation.
 

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