Editing Fermi gas (section)

=== Thermodynamic limit ===
When the box contains ''N'' non-interacting fermions of spin-{{sfrac|1|2}}, it is interesting to calculate the energy in the thermodynamic limit, where ''N'' is so large that the quantum numbers ''n''<sub>''x''</sub>, ''n''<sub>''y''</sub>, ''n''<sub>''z''</sub> can be treated as continuous variables.

With the vector <math>\mathbf{n}=(n_x,n_y,n_z)</math>, each quantum state corresponds to a point in 'n-space' with energy
<math display="block">E_{\mathbf{n}} = E_0 + \frac{\hbar^2 \pi^2}{2m L^2} |\mathbf{n}|^2 \,</math>

With <math> |\mathbf{n}|^2 </math>denoting the square of the usual Euclidean length <math> |\mathbf{n}|=\sqrt{n_x^2+n_y^2+n_z^2} </math>.
The number of states with energy less than ''E''<sub>F</sub>&nbsp;+ ''E''<sub>0</sub> is equal to the number of states that lie within a sphere of radius <math>|\mathbf{n}_{\mathrm{F}}|</math> in the region of n-space where ''n''<sub>''x''</sub>, ''n''<sub>''y''</sub>, ''n''<sub>''z''</sub> are positive. In the ground state this number equals the number of fermions in the system:

<math display="block">N =2\times\frac{1}{8}\times\frac{4}{3} \pi n_{\mathrm{F}}^3 </math>

[[File:K-space.JPG|thumb|The free fermions that occupy the lowest energy states form a [[sphere]] in [[reciprocal lattice|reciprocal space]]. The surface of this sphere is the [[Fermi surface]].]]
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.
<math display="block">n_{\mathrm{F}}=\left(\frac{3 N}{\pi}\right)^{1/3} </math>
The '''Fermi energy''' is given by
<math display="block">E_{\mathrm{F}} = \frac{\hbar^2 \pi^2}{2m L^2} n_{\mathrm{F}}^2 = \frac{\hbar^2 \pi^2}{2m L^2} \left( \frac{3 N}{\pi} \right)^{2/3}</math>

Which results in a relationship between the Fermi energy and the [[particle number density|number of particles per volume]] (when ''L''<sup>2</sup> is replaced with ''V''<sup>2/3</sup>):
:{{box|border color=#ccccff|<math>E_{\mathrm{F}} = \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N}{V} \right)^{2/3} </math>}}

This is also the energy of the highest-energy particle (the <math>N</math>th particle), above the zero point energy <math>E_0</math>. The <math>N'</math>th particle has an energy of
<math display="block"> E_{N'} = E_0 + \frac{\hbar^2}{2m} \left( \frac{3 \pi^2 N'}{V} \right)^{2/3} \,=E_0 + E_{\mathrm{F}} \big |_{N'} </math>

The total energy of a Fermi sphere of <math>N</math> fermions (which occupy all <math>N</math> energy states within the Fermi sphere) is given by:

<math display="block">E_{\rm T} = N E_0 + \int_0^N E_{\mathrm{F}}\big |_{N'} \, dN' = \left(\frac{3}{5} E_{\mathrm{F}} + E_0\right)N</math>

Therefore, the average energy per particle is given by:
<math display="block"> E_\mathrm{av} = E_0 + \frac{3}{5} E_{\mathrm{F}} </math>