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The position operator (defined within Schroedinger representation as usual)becomes meaningless when the usual Born-von Karman periodic boundary conditionsare adopted: this fact is at the root of the polarization problem. I show howto 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 continuumintegral, and is computationally very useful for disordered systems.Simulations which are based on this concept are being performed by severalgroups.Comment: 10 pages, 1 embedded figure (in two panels). Presented at the Fifth  Williamsburg Workshop on First-Principles Calculations for Ferroelectric

The quantum-mechanical position operator and the polarization problem