Adiabatic theorem for finite dimensional quantum mechanical systems

TL;DR

This paper presents a straightforward proof of the adiabatic theorem for finite-dimensional quantum systems, including degenerate cases, and demonstrates the optimality of the error estimate with an example.

Contribution

It provides a new simple proof of the adiabatic theorem applicable to both nondegenerate and degenerate states in finite-dimensional quantum systems.

Findings

The proof is explicitly integrable for a two-level system.

The error estimate of the adiabatic theorem cannot be improved.

The proof covers both nondegenerate and degenerate quantum states.

Abstract

A new simple proof of the adiabatic theorem is given in the finite dimensional case for nondegenerate as well as degenerate states. The explicitly integrable two level system is considered as an example. It is demonstrated that the error estimate given by the adiabatic theorem can not be improved.