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

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.