On the Lewis-Riesenfeld (Dodonov-Man'ko) invariant method

TL;DR

This paper compares the Lewis-Riesenfeld invariant method with the Quantum Arnold-Ermakov-Pinney transformation for solving quantum time-dependent harmonic oscillators, highlighting the latter's advantages especially in damped cases.

Contribution

It demonstrates the equivalence of the two methods and shows the superiority of the Quantum Arnold-Ermakov-Pinney transformation in certain quantum oscillator problems.

Findings

Quantum Arnold-Ermakov-Pinney transformation offers advantages over Lewis-Riesenfeld method.

The invariant by Dodonov & Man'ko is more suitable for damped oscillators.

Examples illustrate the effectiveness of the proposed approach.

Abstract

We revise the Lewis-Riesenfeld invariant method for solving the quantum time-dependent harmonic oscillator in light of the Quantum Arnold Transformation previously introduced and its recent generalization to the Quantum Arnold-Ermakov-Pinney Transformation. We prove that both methods are equivalent and show the advantages of the Quantum Arnold-Ermakov-Pinney transformation over the Lewis-Riesenfeld invariant method. We show that, in the quantum time-dependent and damped harmonic oscillator, the invariant proposed by Dodonov & Man'ko is more suitable and provide some examples to illustrate it, focusing on the damped case.