Moments of probability density of Hydrogen atom in a cage

Abstract

There are various information theoretic measures to study many complex systems. Most of the attention has been confined to the study of the free and confined hydrogen atom for the ground state. The Shannon entropy, Fisher entropy, Renyi entropy, and Tsalli entropy have been investigated extensively in the literature. These entropies are non-extensive and employ only the first moment of probability distribution functions. In the present work, we have presented results of the confined Hydrogen atom in a hard and a soft wall, in which higher moments of probability distribution functions are used that have a close relationship with entropic moments. These moments are the average values of nth power of probability distribution functions. The $n$ = 2 moments in ($r$-) and ($p$-) spaces are specifically known as Onicescu energies and are denoted by ${E}_{r}$ and ${E}_{p}$. Since the position ($r$-) and the momentum ($p$-) wavefunctions are connected through Fourier transformation, all the moments in ($r$-) and ($p$-) spaces exhibit contrasting behaviour. For large confining radius, these moments attain their asymptotic values of the free Hydrogen atom. In confining environments, the ($r$-) and ($p$-) curves cross each other. The product of each moment in ($r$-) and ($p$-) spaces remain almost constant, in conformity with the Uncertainty product. The effect of confinement is noticeable below the crossing points. We have also evaluated ${E}_{r}$, for a confined attractive short-range spherical Gaussian type potential.