Quantum information entropy for a hyperbolical potential function

The exact solutions to the Schrödinger equation with a hyperbolic potential are obtained. The position S_x and momentum S_p Shannon information entropies for the low-lying states are calculated. Some interesting features of the information entropy densities and as well as the probability densities and are demonstrated. We find that the choices of the values for those parameters have to satisfy the condition on . We also notice that the and are symmetric to the momentum p and the or is equal or greater than 1 at some positions r or momentum p. In addition, the Bialynicki-Birula-Mycielski inequality is tested from different cases and found to hold for these cases.