The quantum-mechanical position operator and the polarization problem

TL;DR

This paper addresses the polarization problem in quantum mechanics by redefining the position expectation value using a many-body operator, providing a computationally useful approach especially for disordered systems.

Contribution

It introduces a novel definition of the position operator suitable for periodic boundary conditions, generalizing previous work and connecting to the Berry phase concept.

Findings

The new position expectation value coincides with the single-point Berry phase for uncorrelated electrons.

The approach is computationally advantageous for disordered systems.

Simulations based on this method are actively being performed by research groups.

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.