Adiabatic Conditions and the Uncertainty Relation

TL;DR

This paper clarifies the physical meaning of adiabatic conditions by linking them to the uncertainty relation, and proposes a new sufficient condition for adiabatic approximation with a clear physical interpretation.

Contribution

It relates adiabatic conditions to the uncertainty principle and introduces a new sufficient condition with a physical understanding.

Findings

The traditional quantitative condition corresponds to the transition probability amplitude.

The transition probability is related to the time uncertainty of the system.

A new sufficient condition for adiabatic approximation is proposed.

Abstract

The condition for adiabatic approximation are of basic importance for the applications of the adiabatic theorem. The traditional quantitative condition was found to be necessary but not sufficient, but we do not know its physical meaning and the reason why it is necessary from the physical point of view. In this work, we relate the adiabatic theorem to the uncertainty relation, and present a clear physical picture of the traditional quantitative condition. It is shown that the quantitative condition is just the amplitude of the probability of transition between two levels in the time interval which is of the order of the time uncertainty of the system. We also present a new sufficient condition with clear physical picture.