⚛️ quantum physics

Position- and Momentum-Space Quantum Information Measures of the Double-Morse Oscillator

This paper analytically investigates the quantum information properties of the double Morse oscillator in position and momentum spaces, revealing how Shannon entropy, Fisher information, and related measures evolve as the potential transitions from a distinct double-well to a single-well profile, with the ground state approaching Gaussian behavior while the excited state retains non-Gaussian characteristics.

Original authors: Firoz Chogle, Ernesto Damiani, Berihu Teklu

Published 2026-08-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Firoz Chogle, Ernesto Damiani, Berihu Teklu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, particles like electrons do not sit still; they exist as clouds of probability, spreading out across space and time. To understand these clouds, scientists have long relied on measuring average values, such as where a particle is most likely to be found or how fast it is moving on average. However, these averages only tell part of the story. They miss the intricate details of the particle's shape and how its probability is organized. To capture this full picture, researchers have turned to information theory, a field originally developed to measure uncertainty in communication. In this context, information measures act as a sophisticated lens, quantifying how spread out a particle is, how concentrated its energy becomes, and how complex its internal structure is. These tools allow scientists to see not just where a particle is, but how "ordered" or "chaotic" its existence is, revealing subtle shifts that standard measurements would overlook.

A team of researchers at Khalifa University and the University of Milan has applied these information-theoretic tools to a specific and fascinating model known as the double Morse oscillator. Imagine a valley with two distinct dips separated by a small hill, where a particle can rest in either dip. This is the "double-well" potential. By adjusting a single control parameter, the researchers could smoothly transform this landscape, causing the two separate dips to merge into a single, wide valley. Their goal was to track how the particle's behavior changes during this transformation, moving from a state where it is trapped in two separate locations to a state where it occupies a single, unified space. They focused on the two lowest energy states of the system—the ground state and the first excited state—because the mathematical model they used allows for precise, analytical solutions for these specific levels.

The researchers calculated several key quantities to describe the particle's behavior in both position (where it is) and momentum (how it is moving). They looked at measures of uncertainty, which tell us how spread out the particle is; measures of concentration, which show how tightly packed the probability is; and measures of complexity, which reveal how much structure exists within the distribution. As they adjusted the parameter that controls the shape of the potential, they observed a clear and complementary shift. When the two wells were far apart, the particle was highly delocalized in position, meaning it was spread out over both dips with equal likelihood. In this state, the particle's position was very uncertain, but its momentum was surprisingly well-defined and concentrated. As the parameter changed and the two wells began to merge into one, the particle became more localized in the center of the new single valley. This caused the position uncertainty to drop, but the momentum distribution to spread out and become more complex.

One of the most significant findings concerns the ground state, the lowest energy level of the system. As the two wells merged completely, the researchers found that the ground state began to behave more like a simple, smooth Gaussian curve, which is the standard shape for a particle in a simple harmonic trap. They measured this by calculating a specific product of information measures, which equals a value of one for a perfect Gaussian shape. As the wells merged, this value for the ground state approached one, indicating a transition toward this simple, smooth behavior. However, the excited state, which has a more complex internal structure with a node or a point of zero probability in the middle, did not follow this trend. Even as the wells merged, the excited state retained a strong, non-Gaussian character, maintaining its structural complexity. This suggests that while the simplest state of the system can smooth out into a basic shape, more complex states resist this simplification, preserving their unique internal features.

The study also highlighted how these information measures can serve as a diagnostic tool for real-world experiments. The parameter used to control the shape of the potential is not just a theoretical number; it corresponds to physical quantities that can be measured in laboratories, such as the height of the barrier between wells, the distance separating them, or the local frequency of vibration. Whether in systems of ultracold atoms trapped by lasers, ions held in electromagnetic fields, or even in the behavior of protons in hydrogen bonds, the same mathematical trends apply. The researchers demonstrated that by measuring these information-theoretic quantities, scientists can gain a deeper understanding of how a system transitions between different regimes of confinement. They showed that while the ground state becomes increasingly simple and Gaussian-like as the potential merges, the excited state remains structurally distinct, offering a clear signature of its complexity. This work provides a new way to quantify the "shape" of quantum states, offering a bridge between abstract mathematical models and the tangible properties of physical systems.

Technical Summary: Position- and Momentum-Space Quantum Information Measures of the Double-Morse Oscillator

Problem Statement
This work investigates the quantum-information properties of a particle confined within a double Morse potential. While conventional observables (moments of position, momentum, and energy) characterize selected averages of a quantum state, they do not fully capture the spatial and momentum probability distributions. The authors aim to fill a gap in the literature by providing a systematic information-theoretic analysis of the double Morse oscillator, a model capable of continuously tuning from a well-separated double-well structure to a merged single-well profile. Previous studies have addressed the optical response, nonlinearity, and metrological properties of this system, but a comprehensive analysis using information-theoretic measures across the double-well-to-single-well crossover was missing.

Methodology
The study focuses on the two analytically accessible bound states (the ground state and the first excited state) of the quasi-exactly solvable double Morse oscillator. The potential is defined as $V_{DM} = D(A \cosh(\alpha x) - 1)^2$ , where the dimensionless parameter $A = 2e^{-\alpha x_0}$ controls the morphology of the potential (ranging from distinct double wells at small $A$ to a merged single well as $A \to 1^-$ ).

The authors utilize the quasi-exact solvability of the model (specifically for the sector where $\mu = n+1$ with $n=1$ ) to derive analytical expressions for the position wavefunctions $\psi(y)$ and their corresponding energies. The momentum-space wavefunctions $\phi(k)$ are obtained via Fourier transformation.

To characterize these states, the authors evaluate five specific information-theoretic measures as functions of the parameter $A$ :

Shannon Entropy ( $S$ ): Quantifies uncertainty and delocalization in position ( $S_\rho$ ) and momentum ( $S_\gamma$ ) spaces.
Onicescu Energy ( $O$ ): Quantifies disequilibrium or concentration of the probability density.
Fisher Information ( $I$ ): Measures local gradients and sensitivity to changes in the distribution.
Statistical Complexity ( $C_{GL}$ ): Combines entropy and disequilibrium to capture simultaneous spreading and structural organization.
Fisher-Shannon Product ( $P$ ): Diagnoses non-Gaussianity and structural complexity by balancing global spreading and local gradient content.

The analysis tracks how these measures evolve as the parameter $A$ varies, effectively mapping the transition from a double-well regime to a single-well regime.

Key Results

Wavefunction Behavior: As $A \to 0^+$ (distinct wells), the ground-state probability density splits into two lobes, and the particle is delocalized over both wells. As $A \to 1^-$ (merged well), the ground state localizes near the center, exhibiting a single peak, while the excited state retains a two-lobe structure that moves closer together.
Entropy Trends: Position entropy increases as $A \to 0^+$ due to the particle's distribution over two separated wells, whereas momentum entropy decreases in this limit as the momentum distribution becomes sharply peaked around $k=0$ . Conversely, as $A \to 1^-$ , position entropy decreases (localization) while momentum entropy increases (spreading). The total entropy remains nearly constant, satisfying the entropic uncertainty relation.
Gaussianity and Complexity:
- As the wells merge ( $A \to 1^-$ ), the ground state approaches Gaussian-like behavior. This is evidenced by the Fisher-Shannon product approaching unity and the Onicescu product approaching the Gaussian benchmark of $1/(2\pi)$ .
- The excited state retains a stronger non-Gaussian structure due to its nodal character, preventing similar saturation toward Gaussian benchmarks.
- Statistical complexity in position space shows a steep decrease followed by an increase for the ground state as wells merge, reflecting the transition from a two-lobe to a single-lobe structure. Momentum-space complexity remains higher than position-space complexity due to the oscillatory nature of momentum wavefunctions.
Fisher Information: The product of Fisher information in conjugate spaces satisfies the lower bound relation. The momentum-space Fisher information is high for small $A$ (sharp peaks) and decreases as the distribution spreads.

Significance and Claims
The paper claims that information-theoretic measures provide a useful diagnostic tool for quantifying complexity in quasi-exactly solvable double-well systems. The study demonstrates how the redistribution of uncertainty, localization, and structural complexity occurs during the double-well-to-single-well crossover.

Specifically, the authors highlight that:

The ground state's approach to Gaussian behavior in the single-well regime can be detected via complementary density-based signatures (Fisher-Shannon product and Onicescu product).
These measures offer a way to characterize the system's response to changes in the effective potential shape, which can be linked to experimental parameters (such as barrier height and well separation) in platforms like ultracold atoms, trapped ions, and semiconductor quantum wells.
The analysis is strictly limited to the ground and first excited states ( $n=1$ ) due to the quasi-exact solvability constraints; the authors explicitly state that their conclusions should not be extrapolated to higher bound states, which would require numerical diagonalization.

The work serves as a theoretical complement to existing studies on the double Morse oscillator, offering a detailed mapping of information-theoretic properties that could support future applications in quantum information and quantum-state engineering.

More like this