Shannon and Fisher entropy measures for a parity-restricted harmonic oscillator
Ye-Jiao Shi, Guo-Hua Sun, Jian Jing, Shi‐Hai Dong · Laser Physics · 2017
Abstract We first present the analytical solutions to the Schrödinger equation with the parity-restricted harmonic oscillator V ( x ) = 1 2 m ω 2 x 2 ( x > 0 ) and then calculate the Shannon information entropies S x n and S p n both in position space and in momentum space. It is interesting to find that the variation of the Shannon information entropy S p n in momentum space is different from our previous studies, i.e. the S p n 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 ρ s ( x ) and ρ s ( p ) are also demonstrated. We find that the Fisher entropy is exactly given by I F = 8 n + 6 , n = 0 , 1 , 2 , … .