Parameter space geometry of the quartic oscillator and the double well potential: Classical and quantum description

TL;DR

This paper explores the geometry of the parameter space of the quartic oscillator and double well potential using quantum and classical methods, revealing insights into state localization and potential behavior.

Contribution

It introduces a semiclassical approach to analyze the parameter space geometry, bridging quantum and classical descriptions of anharmonic oscillators.

Findings

Quantum and semiclassical metrics coincide for positive and negative potentials.

Perturbative methods fail near the vanishing quartic potential region.

Both formalisms identify the transition point for ground state delocalization.

Abstract

We compute both analytically and numerically the geometry of the parameter space of the anharmonic oscillator employing the quantum metric tensor and its scalar curvature. A novel semiclassical treatment based on a Fourier decomposition allows to construct classical analogues of the quantum metric tensor and of the expectation values of the transition matrix elements. A detailed comparison is presented between exact quantum numerical results, a perturbative quantum description and the semiclassical analysis. They are shown to coincide for both positive and negative quadratic potentials, where the potential displays a double well. Although the perturbative method is unable to describe the region where the quartic potential vanishes, it is remarkable that both the perturbative and semiclassical formalisms recognize the negative oscillator parameter at which the ground state starts to be delocalized in two wells.