Quantum particles constrained on a deforming surface

Abstract

We unify Helfrich curvature theory, widely used in biomembrane mechanics, with the quantum confining potential method to derive a generalized Schrödinger equation for electrons on dynamically deforming surfaces. This framework explicitly incorporates curvature-driven geometric potential $V_g^{\prime }$, combining intrinsic curvature effects and extrinsic deformation corrections, to quantify how surface deformations modify electronic energy levels. For cylindrical nanostructures with radial breathing modes, we predict THz-scale shifts in quantum transition frequencies, experimentally detectable via resonance absorption. Our results bridge biofilm mechanics and quantum dynamics, offering design principles for strain-tunable nanoelectronics, such as curvature-sensitive transistors or frequency-adaptive sensors.