Uncertainty relations for modified isotropic harmonic oscillator and Coulomb potentials S.H. Patil a , K.D. Sen b,∗ a Department of Physics, Indian Institute of Technology, Mumbai 400 076, India b School of Chemistry, University of Hyderabad, Hyderabad 500 046, India Communicated by V.M. Agranovich Abstract 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/r 2 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. Keywords: Isotropic harmonic oscillator; H atom; Davidson potential; Kratzer potential; Central potentials in D dimensions; Heisenberg uncertainty relation; Shannon entropy; Fisher information measure 1. Introduction Uncertainty relations are the basic properties of quantum mechanics. In particular, Heisenberg uncertainty principle [1] for the product of the uncertainties in position and momentum, in terms of Planck’s constant, σ x σ p  1 2 ¯ h, (1) σ 2 x = ( x −〈x 〉 ) 2  , σ 2 p = ( p x −〈p x 〉 ) 2  , is an important element of quantum properties. In this case the Gaussian wave functions have the minimum uncertainty prod- uct of ¯ h/2. For extensive numerical tests of Eq. (1) for the central potentials we refer the reader to the published litera- ture [2]. Here it may be observed that the uncertainty product for the bound states in homogeneous power-law potentials, is independent of the strength of the potential. This follows from the dimensionality argument that for this case ¯ h is the only quantity which has the dimension of xp. Recently, an exten- sion of this effect for the eigendensities of the homogeneous potentials has be proposed [3]. An interesting case would be the uncertainty relation for the bound states for a superposition of two power-law potentials, and its dependence on the strengths of the two terms. We consider two special cases of superpositions of poten- tials, the isotropic harmonic oscillator (h.o.) potential with an additional a/r 2 term, (2) V 1 (r) = 1 2 kr 2 + a r 2 and the Coulomb potential with an additional a/r 2 term, (3) V 2 (r) =− Z r + a r 2 . We note here that V 1 (r) represents the modified oscillator potential proposed by Davidson [4] which has been found use- ful in analyzing [5] the roto-vibrational states of diatomic mole- cules. A five-dimensional Davidson model potential has been employed to study the nuclear rotations [6] and vibrations. Fur-