Abstract
The matrix representations for the space translations appropriate to Schrödinger’s equation for a one-dimensional periodic potential are developed. From these matrices the canonical form for the dispersion function relating the energy and the Bloch wave vector is derived. The Lie algebra of the group of the Schrödinger equation is used to identify the corrections made to semiclassical band theory by the noncommutivity of the quantum-mechanical representations. As an example of the usefulness of the representational approach, the Saxon–Hutner–Luttinger theorem for bands in composite lattices is proved in general, and extended to the whole dispersion function, and for any number of inequivalent atoms. The proof is based on an addition theorem for Bloch wave vectors of inequivalent lattices.
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