Non-Relativistic Quantum Particle Confined on a Cylindrical Surface under a Stark-like Potential

Abstract

This study explores the influence of a Stark-like perturbative potential on a quantum particle confined to a cylindrical surface (QPCS) and its implications for extra-dimensional theories. The QPCS framework is particularly relevant to Kaluza-Klein (KK) theory, which postulates extra spatial dimensions to unify electromagnetism and gravity. In KK theory, these extra dimensions are typically hidden and require high-energy conditions for detection. Motivated by the challenge of uncovering these dimensions more feasibly, this research applies a perturbative potential of the form \hat{H}_{\text{SL}} = \beta zV_{o_{z}}(\theta) to a QPCS characterized by length \textit{L} and radius R_{o}. This potential is inspired by the Stark effect in hydrogen atoms, where energy level splitting serves as an indicator of an external influence. The study demonstrates that, for a degenerate configuration (R_{o} = \frac{L}{\pi}), the Stark-like perturbation effectively induces energy level splitting, which can be interpreted as a means of revealing hidden dimensions. The first-order energy correction in this scenario depends explicitly on the quantum numbers n_{z} and n_{\theta}, highlighting the potential for this approach to probe extra-dimensional effects in lower-energy quantum systems.