Note on the Existence of Hydrogen Atoms in Higher Dimensional Euclidean Spaces

TL;DR

This paper investigates the theoretical possibility of hydrogen atoms existing in higher-dimensional Euclidean spaces (D > 3) by solving the Schrödinger equation numerically, suggesting such atoms could indeed exist in these spaces.

Contribution

It extends the Schrödinger equation to higher dimensions with a generalized Coulomb potential and demonstrates the potential existence of bound hydrogen states beyond three dimensions.

Findings

Hydrogen atoms can theoretically exist in dimensions 5 to 10.

Energy eigenvalues for higher-dimensional hydrogen states are computed.

Most probable electron-nucleus distances are determined for various dimensions.

Abstract

The question of whether hydrogen atoms can exist or not in spaces with a number of dimensions greater than 3 is revisited, considering higher dimensional Euclidean spaces. Previous results which lead to different answers to this question are briefly reviewed. The scenario where not only the kinematical term of Schr\"odinger equation is generalized to a D-dimensional space but also the electric charge conservation law (expressed here by the Poisson law) should actually remains valid is assumed. In this case, the potential energy in the Schr\"odinger equation goes like 1/r^{D-2}. The lowest quantum mechanical bound states and the corresponding wave functions are determined by applying the Numerov numerical method to solve Schr\"odinger's eigenvalue equation. States for different angular momentum quantum number (l = 0; 1) and dimensionality (5 \leq D \leq 10) are considered. One is lead to the result that hydrogen atoms in higher dimensions could actually exist. For the same range of the dimensionality D, the energy eigenvalues and wave functions are determined for l = 1. The most probable distance between the electron and the nucleus are then computed as a function of D showing the possibility of tiny bound states.