Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well

Abstract

The Shannon information entropy for the Schrödinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum Sp information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a, while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ(x) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs(x) and ρs(p) are demonstrated. The Bialynicki–Birula–Mycielski (BBM) inequality is also tested for these states and found to hold.