As shown by Bloch's theorem, introducing a periodic potential into the Schrödinger equation results in a wave function of the form
A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:
In any perturbation analysis, one must consider the base case to which the perturbation is applied. Here, the base case is with U(x) = 0, and therefore all the Fourier coefficients of the potential are also zero. In this case the central equation reduces to the form
The approximation resulting from the assumption that electrons in metals can be analysed using the kinetic theory of gases, without taking the periodic potential of the metal into account. This approximation gives a good qualitative account of some properties of metals, such as their electrical conductivity. At very low temperatures it is necessary to use quantum statistical mechanics rather than classical statistical mechanics. The free-electron approximation does not, however, give an adequate quantitative description of the properties of metals. It can be improved by the nearly free electron approximation, in which the periodic potential is treated as a perturbation on the free electrons.
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