Classical and quantum behavior of the harmonic and the quartic oscillators

TL;DR

This paper explores the classical and quantum dynamics of harmonic and quartic oscillators using a moments-based formalism, revealing insights into their stationary states and evolution without solving differential equations.

Contribution

It applies a moments formalism to analyze classical and quantum oscillators, including anharmonic systems, providing analytical results for stationary states and dynamics.

Findings

Analytical stationary states for harmonic and quartic oscillators

Differences in classical and quantum evolution of dynamical states

Formalism avoids solving differential equations for these systems

Abstract

In a previous paper a formalism to analyze the dynamical evolution of classical and quantum probability distributions in terms of their moments was presented. Here the application of this formalism to the system of a particle moving on a potential is considered in order to derive physical implications about the classical limit of a quantum system. The complete set of harmonic potentials is considered, which includes the particle under a uniform force, as well as the harmonic and the inverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quartic oscillator is analyzed. Classical and quantum moments corresponding to stationary states of these systems are analytically obtained without solving any differential equation. Finally, dynamical states are also considered in order to study the differences between their classical and quantum evolution.