The volume of the positive quadrant of the Fermi sphere is just the volume of a sphere of radius p, that is:

1 4πp^{3}8 3

The volume of an incremental shell from radius p to p+dp (shown in red) is the differential of this, πp^{2}dp/2. This shell can hold:

πp^{2}dp2(h/2L) ^{3}states or

8πp^{2}dp(h/L) ^{3}electrons.

In unit volume (L=1) this becomes 8π^{2}dp/h^{3} per unit volume.

From E=p^{2}/2m we find:

p^{2}= 2mE so p = √(2mE) and

dp = √(2m) dE2√E