Periodicity and the Bloch theorem

If not for the fact that most solids condense into periodic crystal structures, the field of condensed matter would not exist. The periodicity makes it possible to perform calculations for only one piece of the bulk matter (called a unit cell, see Fig.(1.2), or if you study properties with periodicities longer than a unit cell supercell), which then yields the solution for the whole sample. In a macroscopic sample you have both surface atoms and bulk atoms. The number of surface atoms are of the order of N, or about 1 out of 108 in a macroscopic sample. They are therefore neglected in the calculation of bulk properties, and only included if you want to study specifically some property with regard to surfaces. There are also always defects and impurities present in a sample, but these are, although interesting, neglected in the following discussion (and furthermore for the rest of 

It will be derived below, after a short discussion of what the periodicity means in terms of Fourier expansions to what is called the reciprocal lattice. 

First we define the lattice vectors as the set of vectors a that spans a unit cell of the lattice. Then we define a translation operator T as any (sequence of) translation(s) that leaves the lattice invariant  

Suppose now that we have a periodic function defined for the crystal, so that  

Since the lattice is periodic, so is the external potential that enters the Schrodinger equation, and therefore we can state that because of the translational symmetry  

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