Density of States for the Electron Gas

We will now calculate the density of electron states in the case of the electron gas. In this model, the core electrons are considered as nearly localized, and must be distinguished from the conduction electrons, which are supposed to freely move in Bloch states throughout the whole crystal [5], Because of the fact that the potential is constant, the single-particle Hamiltonian is merely the kinetic energy of the electron, that is, 

the conduction electron states are plane waves, that is, 

the real wave function must include the spin coordinate, then [6] 

The Physical Chemistry of Materials Energy and Environmental Applications 

Substituting Equation 1.38 in Equation 1.36, we will get the energy of the electrons that is independent of the spin state [12,15] 

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