text: 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: