Shannon information entropy for a hyperbolic double‐well potential

We use the ansatz method to obtain the symmetric and antisymmetric solutions of a hyperbolic double‐well potential by solving the Heun differential equation. The Shannon entropy is studied. The position S x and momentum S p information entropies for the low‐lying two states N = 1, 2 are calculated. Some interesting features of the information entropy densities ρ s ( x ) and ρ s ( p ) as well as the probability density ρ ( x ) are demonstrated. We find that ρ ( x ) is equal or greater than 1 at positions for the allowed potential‐depth values of U 0 = 595.84 (symmetric case) and U 0 = 1092.8 (antisymmetric case). This arises from the fact that most of the density is less than 1, the curve has to rise higher than 1 to have a total area of 1 as required for all probability distributions. We find that the position information entropy S x decreases with the potential strength but the momentum entropy S p is contrary to the S x . The Bialynicki‐Birula–Mycielski inequality is also tested and found to hold for these cases.