Quantum Model of Free Electrons in Crystal

A crystal is composed from atoms or assemblies of atoms spatially ordered. The atoms are occupied by electrons in quantum shells, according to the rules of minimum energy occupancy and the maximum orbital penetration which confer them a classification after the complete occupied and the free shells in Periodical Table of associated elements. 

quantum atom in any state (fundamental or excited) can by simplified treated as being composed by a core state that includes the occupied shells (of minimum energy and those orbital close to the nuclei) and a valence state with the farthest (complete or incomplete) shell from the nuclei influence. In quantum terms such a states separation requires the associated eigen-functions to be orthogonal 

Therefore, since the crystal is composed by bonded periodically atoms it can be, in turn, characterized by two basic orthogonal states, namely that of core and that of valence, being also characterized by the wave vector associated to the wave packets of those states, according to the relation (3.53), and intimate bonds (described in previous section) with crystallographic planes by reciprocal lattice and Brillouin zone(s). 

This kind of treatment is based by the so-called pseudo potential theory, wherefrom, for the present case, the free electrons picture inside the crystal which are moving in a potential hole (of length L, crystal length) accordingly results. 

The Schrbdinger equation (3.10) for modeling free electrons in crystal (Putz, 2006)  

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