Quantum information entropies for position-dependent mass Schrödinger problem

Abstract

The Shannon entropy for the position-dependent Schrödinger equation for a particle with a nonuniform solitonic mass density is evaluated in the case of a trivial null potential. The position Sx and momentum Sp information entropies for the three lowest-lying states are calculated. In particular, for these states, we are able to derive analytical solutions for the Sx entropy as well as for the Fourier transformed wave functions, while the Sp quantity is calculated numerically. We notice the behavior of the Sx entropy, namely, it decreases as the mass barrier width narrows and becomes negative beyond a particular width. The negative Shannon entropy exists for the probability densities that are highly localized. The mass barrier determines the stability of the system. The dependence of Sp on the width is contrary to the one for Sx. Some interesting features of the information entropy densities ρs(x) and ρs(p) are demonstrated. In addition, the Bialynicki-Birula–Mycielski (BBM) inequality is tested for a number of states and found to hold for all the cases.