The quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation

TL;DR

This paper proves that the commonly used quantitative condition is necessary for ensuring the validity of the adiabatic approximation, clarifying a long-standing theoretical question.

Contribution

It establishes the necessity of the quantitative condition for the adiabatic approximation, resolving a previously unresolved issue.

Findings

The quantitative condition is necessary for the adiabatic approximation.

It clarifies the theoretical foundation of the adiabatic theorem.

Provides a rigorous proof of the condition's necessity.

Abstract

The usual quantitative condition has been widely used in the practical applications of the adiabatic theorem. However, it had never been proved to be sufficient or necessary before. It was only recently found that the quantitative condition is insufficient, but whether it is necessary remains unresolved. In this letter, we prove that the quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation.