A mathematical model of a quantum oscillator shows that two different changes can look similar in phase space but mean different things. Changing the model's geometry changes the state itself. Changing the ordering parameter changes only the way that fixed state is represented and smoothed. The distinction is the central result of a preprint on the lowest quasi-exact ground state of a symmetric double-Morse oscillator.
The calculation also finds that the state's Wigner representation retains negative regions across the modeled double-well range. In the paper's interpretation, those regions mark nonclassical and non-Gaussian structure. They are not, however, a direct measure of how much negativity the state contains, because the plotted changes in lobe size do not establish a monotonic negative volume.
A double well with a centered ground state
The study is entirely theoretical. It uses analytically derived wavefunctions and phase-space functions rather than participants, experiments or an empirical control group. The construction selects the lowest quasi-exact sector, where the sector label is zero and the fixed dimensionless parameter mu is one. That choice gives a normalizable, node-free ground state for the model.
For values of the geometry parameter A strictly between zero and one, the potential has two wells. Yet the exact ground-state amplitude is single-peaked at the origin and sits above the internal barrier. In other words, the double-well shape does not force this particular ground state to split into two separate peaks.
As A approaches one from below, the two minima move toward the origin and the barrier disappears. The resulting single well remains anharmonic rather than becoming the familiar harmonic oscillator. This limit matters because it shows that the model's geometry changes continuously without reducing the ground state to a purely harmonic case.
Sharper position structure, broader momentum structure
The plotted comparisons show a reciprocal shift between position and momentum descriptions as A increases. The position-space amplitude becomes narrower, while the momentum-space amplitude becomes broader and its oscillatory side lobes weaken. The figures illustrate this pattern with A set to 0.1, 0.5 and 0.8.
The paper turns that family of wavefunctions into an exact Wigner function, a phase-space representation that records the joint structure of the state in position and momentum. Its negative patches persist throughout the double-well range. The authors read that persistence as a sign of nonclassicality and non-Gaussianity, while cautioning that visual changes in the patches do not by themselves give a numerical negative-volume trend.
One state, several ways to describe it
A second object, the Weyl characteristic function, provides a Fourier-space description of the same state. It is normalized to one at the origin and is real and even in both Fourier variables. The paper uses it to generate symmetrically ordered moments and cumulants, and treats it as the generator of the broader family of s-ordered representations.
Here, s is the Cahill-Glauber ordering parameter, which controls the representation rather than the modeled potential. Starting at the Wigner choice s=0 and moving toward the Husimi endpoint s=-1 applies Gaussian smoothing. Fine negative lobes are progressively suppressed, while broad localization remains, and the endpoint distribution is nonnegative.
That final positivity has to be read carefully. The paper does not treat a nonnegative Husimi distribution as proof that the state is classical. At the opposite endpoint, s=1, the P representation is distributional rather than an ordinary integrable function for this non-Gaussian pure state. The analysis therefore separates positivity in one representation from the broader question of classicality.
An exact benchmark, with a narrow scope
The work's strength is also its boundary. It gives an exact analytical benchmark for one lowest quasi-exact ground-state family and shows how physical parameter changes can be kept separate from representation-dependent smoothing. It does not characterize every bound state or every excited state of the full double-Morse Hamiltonian, and it reports no experiment, participant sample or external validation.
The analysis does not calculate a numerical nonclassical depth, the amount of smoothing needed to remove all nonclassical features. It also does not establish that a quantitative measure of Wigner-negative volume changes monotonically with A. The authors identify higher quasi-exact sectors, continuous scans of s, filtered P representations and dynamical or device-specific extensions as questions for further work.
The preprint says its plot data were generated by directly implementing the equations and parameter values in the text and can be obtained from the corresponding author on reasonable request. The research was funded by Khalifa University of Science and Technology through Project ID KU-INT-RIG-2024-8474000739.
Paper data and sources
Original title: Exact Quasiprobability Hierarchy of the Double-Morse Oscillator: From Potential Geometry to Operator Ordering
Authors: F. Chogle, B. Teklu, M. F. Pereira
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
Original paper · Full text