Product of position and momentum Fisher information measures under homogeneous potentials

Abstract

The Fisher information has proved to be a very useful tool in the informational analysis of atomic and molecular systems. In this paper we show analytically that the net Fisher information measure, defined as the product of Fisher information integrals of the electron density in the position, Ir , and momentum, Ip, spaces generated by λV(r) remains independent of strength of the homogeneous potential, λ. Specifically, IrIp=Ir(λ)Ip(λ). This proof must be viewed in the light of a special characteristic of the homogeneous potentials according to which the scaled densities are the eigendensities of the Hamiltonian of the same form with a modified potential strength constant. Furthermore, the behavior of a new form of Fisher information for eigendensities of homogeneous potentials is explored.