Scaling properties of net information measures for bound states of spherical model potentials confined with finite barrier #

Abstract

Using dimensional analyses, the scaling properties of the Heisenberg uncertainty relationship as well as the various information theoretical uncertainty-like relationships are derived for the bound states corresponding to the superposition of the power potential of the form \(V(r)=Zr^n+\sum_i Z_i r^{n_i}\), where Z, Z i , n, n i are parameters, in the free state as well as in the additional presence of a spherical penetrable boundary wall located at radius R The uncertainty product and all other net information measures are shown here to depend only on the parameters [s i ] defined by the ratios \(Z_i/Z^{(n_i+2)/(n+2)}\). Introduction of a finite potential, V c at the radial distance r ≥ R results in a complete set of scaling parameters given by [s i , t 1, t 2], where t 1 is given by RZ 1/(n + 2) and \(t_2= V_c /(Z)^{2/(n+2)}\). The scaling of the sum of Shannon information entropy, S T , with the nuclear charge, Z, for the hydrogen isoelectronic series confined inside an impenetrable cavity of radius R . The optimum value of S T remains fixed for all Z, while the location of R opt scales as 1/Z.