Angelina N. Mendoza Tavera

arXiv2025

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

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.

quant-ph↗

arXiv2025

Entropic characterization of Tunneling and State Pairing in a Quasi-Exactly Solvable Sextic Potential

We analyze the (de)localization properties of a quasi-exactly solvable (QES) sextic potential $V_{\text{QES}}(x) = \frac{1}{2}(x^6 + 2x^4 - 2(2λ+ 1)x^2)$ as a function of the tunable parameter $λ\in [-\frac{3}{4}, 6]$. For $λ> -\frac{1}{2}$, the potential exhibits a symmetric double-well structure, with tunneling emerging for the ground state level at $λ\approx 0.732953$. {For the lowest energy states $ n = 0,1,2,3 $, we construct physically meaningful variational wavefunctions that $i)$ respect parity symmetry under the transformation $x \rightarrow -x$, $ii)$ exhibit the correct asymptotic behavior at large distances, and $iii)$ allow for exact analytical Fourier transforms. Variational energies match Lagrange Mesh and available exact analytical QES results with relative errors $\simeq 10^{-8}$ for $n = 0, 1, 2$ and $\simeq 10^{-6}$ for the third excited state $n=3$. We demonstrate that entropic measures (Shannon entropy, Kullback-Leibler, and Cumulative Residual Jeffreys divergences) surpass conventional variance-based methods in revealing tunneling transitions, wavefunction symmetry breaking, and quantum state pairing. Our results confirm that the Beckner-Bialynicki-Birula-Mycielski entropic uncertainty relation holds across all examined values of $n$ and $λ$. The quality of the trial function is also validated by the small $\sim10^{-10}$ Cumulative Residual Jeffreys divergences from the exact QES solutions.

quant-ph↗