How to Compute Electron Count and Energy at Zero Temperature?

[helpcometk]

Homework Statement

Consider an electron gas with a density of states given by D(e) = ae2. Here a is a constant. The Fermi energy is e_F.
a) We first consider the system at zero temperature. Compute the total number of electrons N and the groundstate
energy E. Show that the average energy per electron in the groundstate is given by (3/4)e_F.

Homework Equations

many available expressions for the number of electrons but don't know which one to use.

i.e N(E)= V/3π²(2mE/hbar²)^3/2

lacking expressions for energy

The Attempt at a Solution

 

[Thaakisfox]

What does the "density of states" mean? it means the number of states between energies e and e+de.

At zero temperature all the states are occupied till the fermi level. So you simply have to integrate the density of states till the Fermi energy to obtain the number of electrons (As electrons obey pauli exclusion, so only one electron per state...).