The quantum-mechanical position operator and the polarization problem
Abstract
The position operator (defined within Schroedinger representation as usual) becomes meaningless when the usual Born-von Karman periodic boundary conditions are adopted: this fact is at the root of the polarization problem. I show how to define the position expectation value by means of rather peculiar many-body (multiplicative) operator acting on the wavefunction of the extended system. This definition can be regarded as the generalization of a precursor work, apparently unrelated to the polarization problem. For uncorrelated electrons, the present finding coincides with the so-called "single-point Berry phase" formula, which can hardly be regarded as the approximation of a continuum integral, and is computationally very useful for disordered systems. Simulations which are based on this concept are being performed by several groups.
Cite
@article{arxiv.cond-mat/9802004,
title = {The quantum-mechanical position operator and the polarization problem},
author = {R. Resta},
journal= {arXiv preprint arXiv:cond-mat/9802004},
year = {2009}
}
Comments
10 pages, 1 embedded figure (in two panels). Presented at the Fifth Williamsburg Workshop on First-Principles Calculations for Ferroelectrics