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.