Quantum-Classical Correspondence in a Quartic Oscillator -- Corrections to Frequency and Bounds for Periodic Motion

TL;DR

This paper compares quantum and classical quartic oscillators, deriving corrections to their frequencies and bounds for periodic motion using perturbation methods, highlighting their similarities and differences.

Contribution

It introduces a method to compute quantum corrections to classical frequencies and establishes bounds for periodicity in both regimes.

Findings

Quantum corrections to frequency depend on amplitude.

Derived bounds for periodic oscillations in quantum and classical cases.

Quantum behavior mimics classical oscillators with the same Hamiltonian in coherent states.

Abstract

We take a qualitative comparative look at quantum and classical quartic anharmonic oscillators. It has been shown that the behavior of the quantum anharmonic oscillator mimics that of the classical anharmonic oscillators with the structurally same Hamiltonian in the coherent state basis of the harmonic oscillators. The associated equation of motion allows us to use Lindstet-Poincare perturbation method to compute the classical frequency of the oscillation order by order, by taking care of its dependence on amplitude and the quantum corrections. We also derive a bound for periodicity of such oscillations in both the classical and quantum cases.