A Schrödinger equation is solved for a cotangent hyperbolic potential using the elegant parametric Nikiforov–Uvarov method. The results obtained are used to calculate the Fisher information for position space and momentum space as well as the Shannon entropy for position space and momentum space at the ground state. The energies of the cotangent potential are observed to be fully bounded for different quantum states. The Fisher information satisfied the Fisher relation known as Cramer–Rao inequality for the Fisher product. Its result also obeyed the Heisenberg principle as its alternative for uncertainty measurement. However, the Shannon entropy only satisfied the Bialynick-Birula, Mycielski inequality for higher values of the screening parameter. Its relation to the Heisenberg principle failed as both the position space and the momentum space are parallel with high entropy squeezing. This potential has not been reported for any system to the best of our understanding.
A Schrödinger equation is solved for a cotangent hyperbolic potential using the elegant parametric Nikiforov–Uvarov method. The results obtained are used to calculate the Fisher information for position space and momentum space as well as the Shannon entropy for position space and momentum space at the ground state. The energies of the cotangent potential are observed to be fully bounded for different quantum states. The Fisher information satisfied the Fisher relation known as Cramer–Rao inequality for the Fisher product. Its result also obeyed the Heisenberg principle as its alternative for uncertainty measurement. However, the Shannon entropy only satisfied the Bialynick-Birula, Mycielski inequality for higher values of the screening parameter. Its relation to the Heisenberg principle failed as both the position space and the momentum space are parallel with high entropy squeezing. This potential has not been reported for any system to the best of our understanding.
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.
Abstract The Shannon entropy (S) and the Fisher Information (I) entropies are investigated for a generalized hyperbolic potential in position and momentum spaces. First, the Schrodinger equation is solved exactly using the Nikiforov‐Uvarov‐Functional Analysis method to obtain the energy spectra and the corresponding wave function. By Fourier transforming the position space wave function, the corresponding momentum wave function was obtained for the low‐lying states corresponding to the ground and first excited states. The positions and momentum of Shannon entropy and Fisher Information entropies were calculated numerically. Finally, the Bialynicki‐Birula and Mycielski and the Stam‐Cramer‐Rao inequalities for the Shannon entropy and Fisher Information entropies, respectively, were tested and were found to be satisfied for all cases considered.
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.
This work investigates the behavior of Shannon entropy and Fisher information for the Varshni-Hellmann potential (VHP) in one and three dimensions using the Nikiforov-Uvarov method. We employ the Greene-Aldrich approximation scheme to obtain the energy eigenvalues and normalized wavefunctions, which are then used to calculate these information-theoretic quantities. Our analysis revealed remarkably similar high-order features in both position and momentum spaces. Notably, our calculations showed enhanced accuracy in predicting particle localization within position space. Furthermore, the combined position and momentum entropies obeyed the lower and upper bounds established by the Berkner-Bialynicki-Birula-Mycieslki inequality. Additionally, for three-dimensional systems, the Stam-Cramer-Rao inequalities were fulfilled for different eigenstates with respect to the calculated Fisher information. It is observed that as the position Fisher entropy decreases, indicating a more precise measurement of position, the momentum Fisher entropy must increase. This implies that the Fisher information regarding momentum decreases, resulting in a decrease in the precision of momentum measurement. This demonstrates how position and momentum uncertainties complement each other in quantum mechanics. Exploring the balance between position and momentum Fisher entropy reveals a fundamental aspect of the uncertainty principle in quantum mechanics, highlighting the restrictions on measuring certain pairs of conjugate variables simultaneously with high precision.
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.
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.