https://doi.org/10.1140/epjc/s10052-025-13909-4
Regular Article - Theoretical Physics
Schrödinger equation in higher-dimensional curved space: a test for the existence of higher dimensions in the quantum realm
Department of Physics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin 10, Famagusta, North Cyprus, Türkiye
Received:
19
September
2024
Accepted:
6
February
2025
Published online:
19
February
2025
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.
© The Author(s) 2025
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