Considerations about the incompleteness of the Ehrenfest's theorem in quantum mechanics

TL;DR

This paper investigates the limitations of Ehrenfest's theorem in quantum mechanics, demonstrating its incompleteness through theoretical arguments and numerical examples, especially regarding the role of Hamiltonian hermiticity.

Contribution

It provides a detailed analysis showing Ehrenfest's theorem is incomplete and clarifies the conditions under which the macroscopic observable derivatives are justified.

Findings

Ehrenfest's theorem is incomplete in certain quantum systems.

Hermiticity of the Hamiltonian alone does not justify all simplifications.

Numerical example with an electric charge supports the theoretical claims.

Abstract

We describe a study motivated by our interest to examine the incompleteness of the Ehrenfest's theorem in quantum mechanics and to resolve a doubt regarding whether or not the hermiticity of the hamiltonian operator is sufficient to justify a simplification of the expression of the macroscopic-observable time derivative that promotes the one usually found in quantum-mechanics textbooks. The study develops by considering the simple quantum system "particle in one-dimensional box". We propose theoretical arguments to support the incompleteness of the Ehrenfest's theorem in the formulation he gave, in agreement with similar findings already published by a few authors, and corroborate them with the numerical example of an electric charge in an electrostatic field. The contents of this study should be useful to Bachelor and Master students; the style of the discussions is tailored to stimulate, we hope, the student's ability for independent thinking.