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

TL;DR

This paper derives a quantum mechanical equation of motion for a particle constrained on a curved hypersurface, revealing a new curvature-induced force alongside the classical centripetal force, impacting interface physics.

Contribution

It introduces a novel curvature-induced force term in the quantum Heisenberg equation for particles on hypersurfaces, extending classical mechanics insights into quantum regimes.

Findings

Quantum particles experience a curvature-induced force proportional to the Laplacian of mean curvature.

The geometric potential influences the particle's dynamics beyond classical centripetal effects.

Curvature-driven interface evolution is fundamentally affected by this additional force.

Abstract

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