Exact Rényi entropies of D-dimensional harmonic systems

Abstract

The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue per se and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowestlying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and Rényi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a D-dimensional quadratic potential, D > 1) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the R´enyi uncertainty measures of the D-dimensional harmonic system for all ground and excited quantum states directly in terms of D, the potential strength and the hyperquantum numbers.