The general proof of certain fundamental equations in the theory of metallic conduction

In the modern theory of electronic conduction the electrons are considered, when the thermal motion of the lattice is neglected, as moving in a periodic potential with the property V (x + la, y + ma, z + na) = V (x, y, z). The wave equation for an electron in this field is { h2/8π2m ∇2 + EK - V} ψK = 0. Block has shown that this equation has solutions of the form ψK = eiK.R UK(R), where UK has the periodicity of the lattice.