Question

For a free electron gas in a metal, the number of states per unit volume with energies from E to E + dE is given by

n(E)dE = 2π/h3(2m)3/2E1/2 dE

Show that the total energy = 3NEmax/5.

Answer 1

If there are N free electrons in the metal there will be N/2 occupied quantum states at the absolute zero of temperature in accordance with the Fermi Dirac statistics. In Fermi-Dirac statistics at absolute zero, kinetic energy is not zero as would be required if the Boltzmann statistics were assumed.

As N(E)dE gives the number of states per unit volume, in a crystal of volume V, the number of electrons in the range from E to E + dE is