In the thermodynamic limit, the total number of particles N are so large that the quantum number n may be treated as a continuous variable. In this case, the overall number density profile in the box is indeed uniform.
The number of quantum states in the range  is:
Without loss of generality, the zero-point energy is chosen to be zero, with the following result:
Therefore, in the range:
the number of quantum states is:
Here, the degree of degeneracy is:
And the density of states is:
In modern literature, the above  is sometimes also called the "density of states". However,  differs from  by a factor of the system's volume (which is  in this 1D case).
Based on the following formula:
the Fermi energy in the thermodynamic limit can be calculated to be: