Validity of the Adiabatic Approximation

TL;DR

This paper critically examines the standard criterion for the validity of the adiabatic approximation, highlighting its limitations and clarifying the distinction between the adiabatic theorem and approximation.

Contribution

It demonstrates that the common criterion is only sufficient under certain conditions and clarifies the difference between the adiabatic theorem and approximation.

Findings

The standard criterion can be insufficient when the Hamiltonian varies rapidly but slightly.

The criterion aligns with the intuitive notion of slowness only in specific cases.

The paper clarifies the conceptual distinction between the adiabatic theorem and approximation.

Abstract

We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the "standard criterion" for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic {\em theorem} and the adiabatic {\em approximation}, two quite different although closely related ideas.