Schrödinger equation in higher-dimensional curved space: a test for the existence of higher dimensions in the quantum realm

Abstract

By considering the possibility of higher dimensions for nonrelativistic quantum particles, we rederive the Schr & ouml;dinger equation (SE) for such particles in a (d-1)-dimensional curved space embedded within a d-dimensional flat space. This approach generalizes de Costa's formalism, which describes a nonrelativistic quantum particle confined to a two-dimensional curved surface embedded in three-dimensional Euclidean space. The original d-dimensional SE is separated into two parts: a one-dimensional global SE, which includes a confining potential to ensure the particle's wavefunction does not propagate into the extra dimension, and a (d-1)-dimensional local SE. The local equation reveals an induced geometric potential, a distinctive feature arising from the presence of higher dimensions. This provides a hypothetical framework for probing the existence of higher-dimensional spaces. We apply this formalism to curved spaces generated by massive central objects, such as black holes or stars, and specifically revisit the behavior of a quantum particle near the Ellis wormhole.