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arXiv:2608.15413

Quantum Information Analysis in a q-Deformed Deng-Fan Model

P. O. Amadi, Collins O. Edet, A. R. P. Moreirae, P. Kalasuwana, N. Ali, R. Endut, S. A. Aljunid

quant-phSubmitted Aug 15, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

We introduce a $q$-deformed Deng-Fan potential ($q$DFP) model that enables controlled modulation of short-range repulsion and long-range attraction while preserving the equilibrium configuration. The model is solved exactly within the framework of the time-independent Schrödinger equation, yielding closed-form expressions for the energy eigenvalues and wave functions in terms of hypergeometric functions. We show that the deformation parameter $q$ induces non-uniform spectral shifts and a redistribution of bound states. In particular, for $q<1$, the system exhibits spectral compression and enhanced spatial localization. In addition, we investigate the system from an information-theoretic perspective using Shannon entropy, Fisher information, and Fisher--Shannon complexity measures in both position and momentum spaces. The results reveal that the deformation parameter governs the redistribution of quantum information, establishing a direct connection between spatial confinement and momentum delocalization in accordance with the Białynicki-Birula and Mycielski entropic uncertainty principle. Stronger deformation pushes the quantum state further from the minimum-uncertainty configuration, increasing the entropic excess above the BBM bound and reducing the information content about complementary observables, even as position-space localization sharpens. The analysis of entropic and Fisher information densities further shows how the deformation reshapes both the local information content and the structural complexity of the quantum states. We show that in the limit $q\to 1$, the $q$DFP model is reduced to the standard Deng-Fan potential.

Comments
26 pages, 5 figures

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We introduce a qqq-deformed Deng-Fan potential (qqqDFP) model that enables controlled modulation of short-range repulsion and long-range attraction while preserving the equilibrium configuration. The model is solved exactly within the framework of the time-independent Schrödinger equation, yielding closed-form expressions for the energy eigenvalues and wave functions in terms of hypergeometric functions. We show that the deformation parameter qqq induces non-uniform spectral shifts and a redistribution of bound states. In particular, for q<1q<1q<1, the system exhibits spectral compression and enhanced spatial localization. In addition, we investigate the system from an information-theoretic perspective using Shannon entropy, Fisher information, and Fisher--Shannon complexity measures in both position and momentum spaces. The results reveal that the deformation parameter governs the redistribution of quantum information, establishing a direct connection between spatial confinement and momentum delocalization in accordance with the Białynicki-Birula and Mycielski entropic uncertainty principle. Stronger deformation pushes the quantum state further from the minimum-uncertainty configuration, increasing the entropic excess above the BBM bound and reducing the information content about complementary observables, even as position-space localization sharpens. The analysis of entropic and Fisher information densities further shows how the deformation reshapes both the local information content and the structural complexity of the quantum states. We show that in the limit q→1q\to 1q→1, the qqqDFP model is reduced to the standard Deng-Fan potential.
This study presents a comprehensive investigation of quantum information measures for the one-dimensional Ultra Generalized Exponential–Hyperbolic Potential (UGEHP) within the framework of the Schrödinger equation. Analytical solutions for the energy eigenvalues and corresponding wave functions are obtained using the Nikiforov–Uvarov method. These solutions serve as the basis for evaluating Shannon entropy and Fisher information in both position and momentum spaces. The numerical results show that the total Shannon entropy satisfies the Beckner–Białynicki-Birula–Mycielski (BBM) inequality, while the Fisher information adheres to the Stam–Cramér–Rao bounds, thereby confirming consistency with fundamental quantum uncertainty principles. In addition, the energy spectrum exhibits a pronounced dependence on the screening parameter and quantum numbers, indicating the tunable confinement properties of the potential. The combined analysis establishes a clear connection between energy quantization, wavefunction localization, and information-theoretic measures, offering deeper insight into the effectiveness of the UGEHP model in describing complex quantum systems.
The bound state solutions of the Schrödinger equation (SE) under a deformed hyperbolic potential are used in this paper to investigate the correlation between the information content, such as Shannon entropies and Fisher information, and the variance of a quantum system in both momentum and position spaces. The variance was obtained from the expectation moments in conjugated spaces and used to get the uncertainty products, Fisher information products and Shannon entropic sums for different potential parameters. These information measures were observed to vary with the potential parameters and obey their lowest bound inequalities. An increase in the Fisher information leads to a lower uncertainty and less information while decreases in the Fisher information result in higher uncertainties. We proposed a relationship between the Shannon entropies, Fisher information and the variance where an increase in the Fisher information leads to a lower Shannon entropy or a lower spread of the probability distribution and vice versa. The numerical results obtained with the proposed relations agree with the most utilized method in the literature. This current work simplifies the calculation of the Shannon entropies notably for complex potential functions.
Fisher information (FI) and Shannon entropy (SP), as fundamental theoretical constructs, play crucial roles in quantum information, offering significant insights into communication systems. This work addresses the radial Schrödinger equation for a cotangent hyperbolic potential, solved using the parametric Nikiforov–Uvarov method. The derived wave functions are utilized to explore FI and SP within a one-dimensional system. The analysis focuses on the influence of the screening parameter and potential depth on these quantities, examining their behavior for the ground state and the first excited state. The study complies with key theoretical principles, including the Cramer–Rao inequality, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality, and the Heisenberg uncertainty principle. The results shed light on the dynamics of quantum states, emphasizing the relationship between quantum uncertainties and information measures. Significant variations in FI and SP emerge with changes in the screening parameter and potential depth, enhancing the understanding of quantum mechanics in confined systems. The findings also reveal the phenomenon of entropy squeezing, underscoring the complex nature of quantum uncertainties. The model demonstrates realistic behavior, with distinct transition points in SP as a function of potential depth, highlighting unique and intriguing characteristics. These contributions enrich the theoretical framework for investigating quantum systems, providing valuable perspectives on their intricate behaviors.
This study utilizes the Nikiforov-Uvarov method to solve the Schrödinger equation for the class of inversely quadratic Yukawa potential (CIQYP), deriving both the energy equation and the normalized wave function. Shannon entropy and Fisher information in both position and momentum spaces are analyzed for low-energy states using the wave function. The Bialynicki-Birula-Mycielski and Stam-Cramer-Rao inequalities are satisfied for the Shannon and Fisher information entropies, illustrating the complementary uncertainties inherent in position and momentum in quantum mechanics. The study underscores the interplay between position and momentum Fisher entropies, reinforcing the Heisenberg uncertainty principle, which imposes limits on the precise simultaneous measurement of conjugate variables. Eigenvalues of the CIQYP for three diatomic molecules (N₂, O₂, and NO) are obtained using their respective data, revealing that the bound state energy spectra of these diatomic molecules increase as both the principal quantum number and angular momentum quantum number rise. Expectation values were numerically determined, and the potential model simplifies to the Kratzer potential under specific boundary conditions, thereby ensuring analytical accuracy. The energy spectra of diatomic molecules such as I₂ and CO are examined, showing that for a fixed principal quantum number, the energy spectrum increases with increasing angular momentum quantum number, in very good agreement with previously obtained results using different analytical methods.
This study explores the influence of deformation parameters on quantum information measures using the q-deformed Hulthén-quadratic exponential-type potential. By solving the Schrödinger equation with the parametric Nikiforov-Uvarov (pNU) method, the energy spectrum and wavefunctions were derived. Quantum information measures, including Shannon entropy and Fisher information, were analyzed for ground states in position and momentum spaces. The results validate the Berkner, Bialynicki-Birula, and Mycieslki, and Stam-Cramér-Rao inequalities, underscoring their consistency with quantum principles. These findings deepen our understanding of deformation effects on quantum systems and offer potential applications in astrophysics and quantum chemistry, advancing the exploration of molecular systems and quantum information theory.
The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.