Entropic Integrals of Hyperspherical Harmonics and Spatial Entropy of D-Dimensional Central Potentials

The information entropy of a single particle in a quantum-mechanicalD-dimensional central potential is separated in two parts. One depends only on the specific form of the potential (radial entropy) and the other depends on the angular distribution (spatial entropy). The latter is given by an entropic-like integral of the hyperspherical harmonics, which is expressed in terms of the entropy of the Gegenbauer polynomials. This entropy is expressed in terms of the values of the quadratic logarithmic potential of Gegenbauer polynomials Cλ n(t) at the zeros of these polynomials. Then this potential for integer λ is given as a finite expansion of Chebyshev polynomials of even order, whose coefficients are shown to be Wilson polynomials.