How to generalize the Ehrenfest theorem and the uncertainty principle

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.