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

By considering the possibility of higher dimensions for nonrelativistic quantum particles, we rederive the Schrödinger 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.