with D ( ϵ ϵ ) = { m ℏ ℏ 2 π π 2 2 m ϵ ϵ ℏ ℏ 2 if ϵ ϵ > 0 , 0 if ϵ ϵ < 0 which is known as the density of levels or density of states per unit volume such that D ( ϵ ϵ ) d ϵ ϵ is the total number of states between ϵ ϵ and ϵ ϵ + d ϵ ϵ . Using the expressions above the integrals can be rewritten as: