Similarity among quantum-mechanical states: analysis and applications for central potentials

Abstract The similarity of quantum-mechanical solutions for central potentials is analytically determined and numerically explored for arbitrary dimensionalities. The study here provided focuses on hydrogenic systems and the harmonic oscillator, in respective non-relativistic frameworks. A diversity of analytical expressions for the quantum similarity measure (QSM) and index (QSI) are provided. Relevant conclusions are derived from the analyses grounded on state quantum numbers, space dimensionality and on the role played by the main characteristic parameters of these systems, namely the nuclear charge in the hydrogenic case, and the angular frequency for the oscillator. For this purpose, a statistical analysis of the QSI values has been performed for a large number both of states and combinations of them in each system. Considering the factorization of QSI into a radial and an angular part, particular attention is paid to the individual contribution of each part in both systems.