Bouncing Wave Packets, Ehrenfest Theorem, and Uncertainty Relation based upon a new Concept for the Momentum of a Particle in a Box

TL;DR

This paper introduces a new self-adjoint momentum operator for a particle in a box, restoring the validity of Ehrenfest's theorem and providing insights into wave packet dynamics and uncertainty relations.

Contribution

It proposes a novel concept for a self-adjoint momentum operator in a box, ensuring Ehrenfest's theorem holds and clarifying the physical meaning of uncertainty relations.

Findings

Ehrenfest theorem is satisfied with the new momentum operator.

Wave packets exhibit spreading, shrinking, and revival behaviors.

A simple form of the uncertainty relation is derived and interpreted.

Abstract

For a particle in a box, the operator $-i\partial_x$ is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg-Robertson-Schr\"odinger uncertainty relation and show that our construction also provides a physical interpretation for it.