I have a question about Fermi energy and the single particle state. I have it a bit hard, on how to formulate my question, for the below text, because I don't fully understand the concepts so well. The text from wiki is:

"The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. In a Fermi gas, the lowest occupied state is taken to have zero kinetic energy, whereas in a metal, the lowest occupied state is typically taken to mean the bottom of the conduction band."

"single-particle state" = Is the state in this case the combination of the quantum number $n, l, m, s$? "in a quantum system of" = would the quantum system be an atom of the material that we are observing, or is the whole material?

If the system is the whole body made of out a specific material, then what would constitute a state? For as much as I know, even though we say electrons in metals are "free" to move in the spaces between, they are actually bound to their respective atoms. Dumb example an electron of an atom is located on $n=3$, $l = p$, $m = -1$, $s = 1/2$.

I know what a conductive band is, but how is does this concept makes sense?

Let's say we have a piece of metal exposed on $T = 0$. Then all the electrons will fall into the lowest possible energy levels for each atom. Is the conductive band, the totality of all electrons of all atoms that are located on the highest $n, l, m, s$ values for each individual atoms?

I hope I somehow made a bit of sense!


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