Accuracy of the quantum adiabatic theorem in its original form

TL;DR

This paper provides a rigorous proof confirming the accuracy of the original quantum adiabatic theorem, clarifies conditions for its violation, and discusses physical implications and comparisons with other proofs.

Contribution

It offers an explicit proof of the quantum adiabatic theorem's accuracy and analyzes conditions leading to its violation, addressing inconsistencies in prior claims.

Findings

The theorem holds under the original adiabatic criterion.

Violations occur only if the adiabatic criterion is broken or degeneracy appears.

Comparison shows the proof's consistency with recent results.

Abstract

An explicit proof is developed to reinforce the accuracy of the quantum adiabatic theorem in its original form without any inconsistency and/or violation. Based on this proof, we discuss physical implications that give rise to the violation of the quantum adiabatic approximation. We show that such a violation can be obtained if and only if one violates the adiabatic criterion itself or due to the existence of degeneracy at a later time. Subsequently, comparison of our proof with respect to other recently developed proofs and counter-examples are analyzed and discussed.