Breakdown of Ehrenfest theorem for free particle constrained on a hypersurface

TL;DR

This paper shows that the Ehrenfest theorem, which relates quantum and classical mechanics, fails for a free particle constrained on a curved hypersurface, challenging its universal applicability.

Contribution

It demonstrates the breakdown of the Ehrenfest theorem for particles constrained on curved hypersurfaces, revealing limitations of the theorem in such geometrical settings.

Findings

Ehrenfest theorem does not hold on curved hypersurfaces

Breakdown occurs for particles on embedded curved surfaces

Challenges the universality of the theorem in constrained systems

Abstract

There is a belief that the Ehrenfest theorem holds true universally. We demonstrate that for a classically nonrelativistic particle constrained on an $N-1$ ($N\geq 2$) curved hypersurface embedded in $N$ flat space, the theorem breaks down.