Relation between position and quasi-momentum operators in band theory

2014 The difference between the position operator and the conjugate of the quasi-momentum operator is expressed in terms of Wannier functions. This provides a simple approach to polarization and to electron dynamics in crystals. J. Phys. France 50 (1989) 2629-2635 15 SEPTEMBRE 1989, Classification Physics Abstracts 71.55C 77.20 03.65 The use of Wannier functions to describe electronic phenomena in crystals is often enlightening [1,2]. They are not only a convenient representation of Bloch waves especially suited to the tightbinding limit, but can also provide natural interpretations for many properties since they constitute the extension to crystal structures of the concept of localized orbitals. The purpose of this note, mainly tutorial, is to show how they can shed light on the partial conjugacy existing in band theory between the position and the quasi-momentum and thus be useful in applications. We recall that, for simple bands, Wannier functions wb(r) are defined as Fourier transforms of Bloch waves PbK.(r) with respect to K, : The integral on the quasi-momentum x (which is defined within translations P of the reciprocal lattice) runs in the Brillouin zone, n denotes the volume of the primitive cell (we let h = 1). The two sets of vectors cpbK.(r) == (rlbK) and wb(r R) r 6R) , labelled by the band index b and by either the quasi-momentum x or the vector R of the Bravais lattice, constitute orthonormal bases exhibiting the translational invariance of the crystal. Provided the phases of the Bloch functions are suitably chosen, the Wannier function wb(r) defined by (1) is localized near the origin (usually Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019890050018026290