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In classical mechanics, a nonrelativistic particle constrained on an $N-1$curved hypersurface embedded in $N$ flat space experiences the centripetalforce only. In quantum mechanics, the situation is totally different for thepresence of the geometric potential. We demonstrate that the motion of thequantum particle is "driven" by not only the the centripetal force, but also acurvature induced force proportional to the Laplacian of the mean curvature,which is fundamental in the interface physics, causing curvature driveninterface evolution.

Heisenberg equation for a nonrelativistic particle on a hypersurface: from the centripetal force to a curvature induced force