When the box contains N non-interacting fermions of spin ½, it is interesting to calculate the energy in the thermodynamic limit, where N is so large that the quantum numbers n, n, n can be treated as continuous variables.
With the vector , each quantum state corresponds to a point in n-space with energy
With denoting the square of the usual Euclidean length .
The number of states with energy less than E +  E is equal to the number of states that lie within a sphere of radius  in the region of n-space where n, n, n are positive. In the ground state this number equals the number of fermions in the system:
The factor of two expresses the two spin states, and the factor of 1/8 expresses the fraction of the sphere that lies in the region where all n are positive.
The Fermi energy is given by
Which results in a relationship between the Fermi energy and the number of particles per volume (when L is replaced with V):
This is also the energy of the highest-energy particle (the th particle), above the zero point energy . The th particle has an energy of
The total energy of a Fermi sphere of  fermions (which occupy all  energy states within the Fermi sphere) is given by:
Therefore, the average energy per particle is given by: