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

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