Free electron gas in one dimensional box

In order to explain experimental facts which were not not satisfactory described by free electron theory of metals, Sommerfield devised the quantum mechanical treatment for those drawbacks of aforesaid theory. He used the Fermi-Dirac distribution and Pauli's exclusion principal instead of Maxwell-Boltzmann distribution. Here free electrons in the metals are assumed as free electron gas or Fermi gas, which are postulated to be inside the one dimensional potential box.