How to generalize the Ehrenfest theorem and the uncertainty principle

TL;DR

This paper introduces generalized forms of the Ehrenfest theorem and the uncertainty principle that remain valid in cases involving azimuthal angles in polar or spherical coordinates, addressing limitations of traditional formulations.

Contribution

It proposes a generalized Ehrenfest theorem and uncertainty relation using the expectation commutator, extending their applicability to problematic coordinate cases.

Findings

Generalized equations are valid for azimuthal angle cases.

The expectation commutator is introduced as a key mathematical tool.

Addresses limitations of traditional quantum mechanical relations.

Abstract

The Ehrenfest theorem and the Robertson uncertainty relation are well-known basic equations in quantum mechanics. However, there exist problematic cases, where the Ehrenfest theorem and the Robertson uncertainty relation are not correct. These cases occur when the azimuthal angle in polar or spherical coordinates is used within these equations. The purpose of this paper is to present and discuss a generalized Ehrenfest theorem and a generalized uncertainty relation which are still valid for these cases. Hereby, we define and use a mathematical operation called expectation commutator.