Quantum information entropies for position-dependent mass Schrödinger problem

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 SxSx and momentum SpSp information entropies for the three lowest-lying states are calculated. In particular, for these states, we are able to derive analytical solutions for the SxSx entropy as well as for the Fourier transformed wave functions, while the SpSp quantity is calculated numerically. We notice the behavior of the SxSx 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 SpSp on the width is contrary to the one for SxSx. Some interesting features of the information entropy densities ρs(x)ρs(x) and ρs(p)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.