The Quantum-Mechanical Position Operator in Extended Systems

TL;DR

This paper proposes a new way to define the position operator in extended quantum systems with periodic boundary conditions, linking it to the Berry-phase theory of polarization.

Contribution

It introduces a many-body operator to define the position expectation value in systems where the standard operator is ill-defined, bridging a gap in quantum condensed matter theory.

Findings

Defined a many-body position operator suitable for periodic systems

Connected the operator to Berry-phase polarization theory

Provided a consistent framework for position expectation in extended systems

Abstract

The position operator (defined within the Schroedinger representation in the standard way) becomes meaningless when periodic boundary conditions are adopted for the wavefunction, as usual in condensed matter physics. We show how to define the position expectation value by means of a simple many-body operator acting on the wavefunction of the extended system. The relationships of the present findings to the Berry-phase theory of polarization are discussed.