Probability Density Operator and Darwin Term in ID Spinless Semi-Relativistic System

Abstract

We derive one-electron/hole probability density in ID semi-relativistic system within quantum field approach, considered as an alternative to Foldy-Wouthuysen transformation. We propose to consider single wall semiconductor carbon nanotube as the physical realization of such ID system. Application of Bogoluybov transformation to nanotube two-band Dirac-like k·p Hamiltonian predicts the additional smearing of position probability density as regards common quantum mechanical smearing, described by the square of wave function. It is shown, that under application of external potential, the revealed difference causes arising of non-local Darwin-like terms in the one-component Salpeter equations, describing separately positive and negative energy states. It is shown within proposed approach that the predicted peculiarity of defined probability density is due to unusual behavior of position operator in multi band k·p problem.