# Density of states of Fermi gas derivation

I'm going over this book. While deriving the gensity of states for a gas of fermions the author makes the following argument:

Remember that we are treating the gas as having a set of states that can be occupied by various numbers of its particles. Those states are just the cells of the everyday position–momentum space that the gas can occupy: three spatial dimensions and three momentum dimensions. Counting those states then equates to counting the number of states available to one free particle moving in three spatial dimensions, given the constraint of some total energy $$E$$.

And then he differentiates the formula for number of states of 1 particle in 3d to obtain the density (and multiplies the density by $$2s+1$$ to account for spin).

I do not understand this. We have a gas of, let's say, $$N$$ fermions. Each of these particles has some particular coordinate in it's 6-D (or rather 7-D because 3-D position + 3-D momentum + 1D for spin). Then any combination of coordinates of each particle in their position-momentum space gives a valid state for the whole fermion gas. I don't see where the analogy with a single particle in 3D comes from.

• Mar 6, 2023 at 13:00