Shannon and Fisher entropy measures for a parity-restricted harmonic oscillator

We first present the analytical solutions to the Schrdinger equation with the parity-restricted harmonic oscillator V(x) =1/2m ω²x² (x > 0) and then calculate the Shannon information entropies Sⁿ_x and Sⁿ_p both in position space and in momentum space. It is interesting to find that the variation of the Shannon information entropy Sⁿ_p in momentum space is different from our previous studies, i.e. the Sⁿ_p first increases with the quantum number n and then decreases with the number n. This is a new and abnormal phenomenon and may be explained by the parity-restricted system. The BBM inequality is verified to be saturated, but the sum of the entropies first increases with the number n and then decreases with it. The entropy densities ps(x) and ps( p ) are also demonstrated. We find that the Fisher entropy is exactly given by IF = 8n + 6, n = 0, 1, 2, . . ..