KvN mechanics approach to the time-dependent frequency harmonic oscillator

TL;DR

This paper explores the use of KvN mechanics and Ermakov-Lewis invariants to analyze the time-dependent frequency harmonic oscillator, unifying quantum and classical dynamics through a common mathematical framework.

Contribution

It introduces a novel approach using Ermakov-Lewis invariants within KvN mechanics to study the oscillator's evolution, applicable to both quantum and classical cases.

Findings

Transformation of Liouville operator using Ermakov-Lewis invariant

Analytical and numerical solutions for the Ermakov equation

Unified framework for quantum and classical dynamics

Abstract

Using the Ermakov-Lewis invariants appearing in KvN mechanics, the time-dependent frequency harmonic oscillator is studied. The analysis builds upon the operational dynamical model, from which it is possible to infer quantum or classical dynamics; thus, the mathematical structure governing the evolution will be the same in both cases. The Liouville operator associated with the time-dependent frequency harmonic oscillator can be transformed using an Ermakov-Lewis invariant, which is also time dependent and commutes with itself at any time. Finally, because the solution of the Ermakov equation is involved in the evolution of the classical state vector, we explore some analytical and numerical solutions.