Direct manifestation of Ehrenfest's theorem in the infinite square well model

TL;DR

This paper demonstrates a direct manifestation of Ehrenfest's theorem in the infinite square well by precisely defining the potential energy term using Dirac delta functions, enabling exact calculation of expectation values.

Contribution

The paper introduces a precise formulation of the potential energy term in the infinite square well, allowing direct verification of Ehrenfest's theorem.

Findings

Potential energy term expressed with Dirac delta functions

Expectation values can be calculated exactly

Ehrenfest's theorem confirmed directly in the infinite well

Abstract

Ehrenfest's theorem in the infinite square well is up to now only manifested indirectly. The manifestation of this theorem is first done in the finite square well, and then consider the infinite square well as the limit of the finite well. For a direct manifestation, we need a more precise formula to describe the degree of infiniteness of the divergent potential energy. We show that the potential energy term term, which is the product of the potential energy and the energy eigenfunction, is a well defined function which can be expressed in terms of Dirac delta functions. This means that the infinity in this model is not that vague but has obtained a specification. This results that expectation values can be calculated precisely and Ehrenfest's thereom can be confirmed directly.