On the Spatial Density Matrix for the Centre of Mass of a one dimensional Perfect Gas

Abstract

We examine the reduced density matrix of the centre of mass on position basis considering a one-dimensional system of $N$ non-interacting distinguishable particles in a infinitely deep square potential well. We find a class of pure states of the system for which the off-diagonal elements of the matrix above go to zero as $N$ increases. This property holds too for the state vectors which are factorized in the single particle wave functions. In this last case, if the average energy of each particle is less than a common bound, the diagonal elements are distributed according to the normal law with a mean square deviation which becomes smaller and smaller as $N$ increases towards infinity. Therefore when the state vectors are of the type considered we cannot experience spatial superpositions of the centre of mass and we may conclude that position is a preferred basis for the collective variable.