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In Sommerfeld theory for metals, after determining all of the possible levels for a single electron, one says that we build up a state for a system with $N$ electrons by filling up those levels, beginning from that which has the lower energy to those which have more energy, in a crescent way. In fact, one call this configuration a Fermi sphere.

My question is: once a level is an eigenstate for the Hamiltonian operator of the problem, why can not the electrons be each one at any random level (respecting, of course, Pauli Exclusion's Principle), not necessarily making a Fermi sphere?

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Because the Fermi sphere is lowest energy. – Ron Maimon Aug 8 '12 at 8:02

The Fermi level is supposed to be the highest occupied state at zero temperature. For fermions at zero temperature, they fill up these states with multiplicity one starting with the ground state up to the Fermi level. This is the lowest energy configuration that abides Pauli exclusion.

At positive temperature (or quantum mechanically) the fermions can be in any state.

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