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 Sx and momentum Sp information entropies for the low-lying two states N = 1, 2 are calculated. Some interesting features of the information entropy densities ρs(x) and ρs(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 Sx decreases with the potential strength but the momentum entropy Sp is contrary to the Sx. The Bialynicki-Birula-Mycielski inequality is also tested and found to hold for these cases.