Still learning about space dimensionality: from the description of hydrogen atom by a generalized wave equation for dimensions D$\geq$3

TL;DR

This paper explores a generalized wave equation for the hydrogen atom in spaces of dimension D≥3, revealing insights into how the dimensionality of space influences the atom's ground state energy and overall behavior.

Contribution

It introduces a generalized Schrödinger equation with an iterated Laplacian and Coulomb-like potential, providing new understanding of space dimensionality's role in atomic properties.

Findings

Ground state energy sign linked to space dimensionality

Ground state energy value related to threefold nature of space

New insights into Ehrenfest's dimensionality hypothesis

Abstract

The hydrogen atom is supposed to be described by a generalization of Schr\"{o}dinger equation, in which the Hamiltonian depends on an iterated Laplacian and a Coulomb-like potential $r^{-\beta}$. Starting from previously obtained solutions for this equation using the $1/N$ expansion method, it is shown that new light can be shed on the problem of understanding the dimensionality of the world as proposed by Paul Ehrenfest. A surprisingly new result is obtained. Indeed, for the first time, we can understand that not only the sign of energy but also the value of the ground state energy of hydrogen atom is related to the threefold nature of space.