Equivalence of the velocity and length gauge perturbation series

TL;DR

This paper proves that the velocity and length gauge perturbation series in quantum mechanics are equivalent representations of the same transition amplitude, belonging to a broader family of solutions, clarifying their relationship.

Contribution

The authors derive a master perturbation expansion in the velocity gauge and demonstrate the equivalence of velocity and length gauge series as branches of a single amplitude.

Findings

Velocity and length gauge series are equivalent in a common domain of convergence.

They are two branches of a one-parameter family of solutions.

The derivation provides a unified framework for understanding gauge choices.

Abstract

We derive a "master" perturbation expansion for the quantum transition amplitude in a light field between the field-free initial and final atomic states in the minimal-coupling (MC) "velocity" gauge. The result is used to prove that the traditional "velocity" and "length" gauge perturbation series are equivalent infinite series representations or branches of the same amplitude function, that are equal but in a common domain of convergence (if it exists). More generally, we show that they constitute only two members of a one-parameter family of infinitely many branches of the given transition amplitude.