A theoretical preprint reports a clear trade-off in a model particle as its double-well landscape is varied from separated wells toward a merged well: position-space entropy falls while momentum-space entropy rises. Despite that exchange, the total entropy stays nearly constant across the modeled transition. In the ground state, it comes close to the fundamental limit but does not reach it.
The study examines a particle confined by a double-Morse potential, a mathematical landscape with two wells whose shape changes through a well-merging transition. The manuscript is an arXiv version 1 preprint dated 26 August 2026, and its findings come from analytical and numerical calculations of the model.
What was calculated
Quasi-exact solvability supplied analytical expressions for the first two bound states and their probability densities. The calculation used the n = 1 sector, containing the ground state and the first excited state. That narrow state set matters: the comparison is between two modeled states, not the full bound-state spectrum.
To study momentum, the wavefunctions were Fourier-transformed, and the later information measures were evaluated numerically. The numerical procedure truncated the decaying integrals at a threshold of 10^-6 and then used adaptive quadrature.
The same shift looks different in each space
Shannon entropy is a way of tracking how spread out a probability distribution is. In the model, position entropy was highest toward the separated-well end, while momentum entropy showed the opposite trend. The near-constant total produced a trade-off between the two spaces rather than a shared rise or fall.
A concentration measure called Onicescu energy reversed the pattern. Position-space Onicescu energy increased with the model parameter A, while momentum-space energy decreased; near the separated-well limit, the ground and excited states approached similar values. The product of the two energies stayed below a Gaussian benchmark. As A increased, the ground-state curve moved toward 1/(2π), but the excited state's node prevented comparable saturation.
Statistical complexity added a more uneven pattern. Ground-state position complexity dropped steeply as A rose and then turned upward. The excited state's position complexity changed more modestly. In momentum space, complexity was higher and followed monotonic trends, making the result dependent on both the state and the space being measured.
Fisher information, which responds to fine-scale structure and localization, was high in momentum space at small A and declined as the distributions spread. At the same time, the position-space wavefunctions developed sharper central structure as A increased. The study reports that the Fisher-information product followed the relation specified by its model.
The ground state moved closer to a bell-curve reference
Several measures pointed toward more Gaussian-like ground-state behavior near the merged-well end. At small A, neither marginal distribution was Gaussian, and the momentum-space wavefunctions were more complex. The ground state's Fisher–Shannon excess, a measure of how far that product sits above its Gaussian reference, declined toward zero as A approached 1.
Relative-entropy non-Gaussianity, another measure of departure from a Gaussian state, also decreased with A for both states. It remained larger for the excited state, matching the Onicescu result: the ground state moved toward the reference more closely, while the excited state retained stronger non-Gaussian structure.
The energy levels tell a more complicated story
The energy levels did not simply collapse as the wells merged. They remained separated near A = 1, but became nearly indistinguishable near A = 0. The ground-state energy reached a reported minimum at A = 0.5, while the first excited-state energy decreased as A decreased.
A narrow window into the model
The conclusions stop at the two states available in the n = 1 closed-form sector. Higher bound states require a different analysis, so these patterns should not be read as applying automatically to the rest of the spectrum.
The reported curves describe qualitative trends, and exact effect sizes and uncertainty intervals are not tabulated for them. The Onicescu observation is a model-specific comparison with a Gaussian benchmark, not a proof of a universal bound.
The result is therefore a statement about this model's two-state calculation, rather than a general conclusion about every bound state.
Paper data and sources
Original title: Position- and Momentum-Space Quantum Information Measures of the Double-Morse Oscillator
Authors: Firoz Chogle, Ernesto Damiani, Berihu Teklu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text