Uncertainty relations for modified isotropic harmonic oscillator and Coulomb potentials

The dimensional analyses of the position and momentum variances which define the Heisenberg uncertainty product are carried out for two non-relativistic model central potentials generated by adding a/r2a/r2 term to (i) the isotropic harmonic oscillator, and (ii) the Coulombic hydrogen-like potentials. The uncertainty products are shown to be independent of the scaling of the part (i) and (ii) but are dependent on the strength a of the additional term. The scaling properties are found to be reflected in the entropic uncertainty measure of the Shannon information entropy sum and the Fisher information product. Numerical results are presented in support of the analytic results derived.