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 _{S x} and momentum _{S p} information entropies for the three lowest-lying states are calculated. In particular, for these states, we are able to derive analytical solutions for the _{S x} entropy as well as for the Fourier transformed wave functions, while the _{S p} quantity is calculated numerically. We notice the behavior of the _{S x} 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 _{S p} on the width is contrary to the one for _{S x}. 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.