Wigner-Husimi phase-space structure of quasi-exactly solvable sextic potential

TL;DR

This paper compares Wigner, modulus-Wigner, and Husimi phase-space distributions in a quantum sextic oscillator to evaluate their effectiveness in capturing interference and structural changes during well separation.

Contribution

It introduces a quantitative hierarchy of phase-space representations' ability to resolve quantum state structures in systems with bimodality or tunneling.

Findings

Wigner function uniquely detects interference effects.

Wigner shows nonmonotonic entropic behavior during well separation.

Husimi and modulus-Wigner mainly reflect geometric splitting.

Abstract

In this study, we compare the Wigner function $W$, its modulus, and the Husimi distribution $H$ in a one-dimensional quantum system exhibiting a transition from a single-well to a double-well configuration, using the quasi-exactly solvable sextic oscillator as a representative example. High-accuracy variational wavefunctions for the lowest states are used to compute two-dimensional phase-space structures, one-dimensional marginals, and the corresponding Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences. The analysis shows that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate, whereas the modulus-Wigner and Husimi distributions account only for geometric splitting or coarse-grained delocalization. These findings establish a quantitative hierarchy in the ability of $W$, $|W|$, and $H$ to resolve structural changes in a quantum state and provide a general framework for assessing the descriptive power of different phase-space representations in systems with emerging bimodality or tunneling.