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 x∼±1.2d for the allowed potential-depth values of U₀ = 595.84 (symmetric case) and U₀ = 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.