The quantum centripetal force on a free particle confined to the surface of a sphere and a cylinder

TL;DR

This paper derives a Hermitian quantum force operator for particles confined to spherical and cylindrical surfaces, revealing a quantum centripetal force component essential for angular momentum and energy conservation.

Contribution

It introduces a geometrically corrected momentum operator and force operator that include a quantum centripetal component, ensuring physical consistency for particles on curved surfaces.

Findings

The force operator has a no-radial component without correction.

The quantum force ensures conservation of angular momentum and energy.

The total force is confirmed to be radial, consistent with classical expectations.

Abstract

The momentum operator for a spin-less particle when confined to a 2D surface embedded into 3D space acquires a geometrical component proportional to the mean curvature that renders it Hermitian. As a consequence, the quantum force operator for a particle confined to spherical and cylindrical surfaces, and free otherwise, derived by applying the Heisenberg equation of motion is found to have an apparently no-radial component in addition to the standard classical radial centripetal force. This component which renders the force operator Hermitian is shown to be essential for the vanishing of the torque the force exerts on the particle and so for the conservation of orbital angular momentum and energy. It is demonstrated that the total force is in fact radial as should be the case for a torque-less one and so can be identified as the quantum centripetal force.