According to quantum mechanics, we can have only certain number of states N per unit volume of phase space in three dimensions, where
|
(6.1) |
Here, Vk3 is the k -space volume and Vr3 is the 3-dimensional space volume. Now assuming that electrons are free to move within the solid in a mean potential –V0 , the energy of the electrons is
|
(6.2) |
where 
is the reduced Planck's constant and me is the mass of electron. If we choose a convenient origin of energy as E = 0, corresponding to the average potential within the metal, then,
|
(6.3) |
Let us now gradually put electrons into the metal. At T = 0, the first two electrons will occupy the lowest energy (or lowest | k |) state and the subsequent electrons will be forced to occupy higher and higher energy states as per Pauli's exclusion principle. Eventually, when all the electrons are accommodated, a sphere of k -space would have been filled up with
|
(6.4) |
Where kF , the Fermi wavevector is the k -space radius of the sphere of filled states. The factor 2 in the above equation denotes the spin degeneracy of the electron. Substituting n = N / Vr3 in the above equation gives,
|
(6.5) |
The corresponding energy is then
|
(6.6) |
