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Physics Open

Information entropies with Varshni-Hellmann potential in higher dimensions

Etido Patrick Inyang, Asmae El Aouami, Norshamsuri Ali, R. Endut, N.R. Ali, Syed Alwee Aljunid

Published May 10, 2024• 8 citations
Publisher

Abstract

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.

DOI
10.1016/j.physo.2024.100220

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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.
This study investigates Fisher and Shannon entropies in one- and three-dimensional systems under the Radial Scalar Power Potential. Using the Nikiforov–Uvarov method combined with the Greene–Aldrich approximation, we derived energy eigenvalues and normalized wavefunctions. The results demonstrate that Shannon and Fisher entropies satisfy fundamental quantum information inequalities, including the Białynicki–Birula–Mycielski and Stam–Cramér–Rao bounds, across different spatial dimensions. Rényi entropy was also analyzed in both position and momentum spaces, revealing its dependence on the screening parameter and highlighting the complementarity in measurement precision between conjugate domains. In particular cases, the Radial Scalar Power Potential reduces to the Kratzer potential, allowing the computation of energy spectra for methylidyne (CH) and nitrogen (N₂) molecules. Energy increases with angular momentum, affecting molecular stability and spectroscopic transitions, while calculated oscillator strengths are in agreement with previous results, thereby validating the Radial Scalar Power Potential model for applications in both quantum information theory and molecular spectroscopy.
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.
Abstract In this paper, Shannon entropy and Fisher information is studied for the screened Kratzer potential model and compared with the screened Coulomb in three dimensions. Our results showed similar higher‐order characteristic behavior for position and momentum space. Our numerical results showed that increases in the accuracy of predicting particle location occurred in the position space. Our result shows that the sum of the position and momentum entropies satisfies the lower‐bound Berkner, Bialynicki‐Birula, and Mycieslki inequality. The Stam‐Cramer‐Rao inequalities relation for Fisher information and the expectation values were also satisfied for the different eigenstates.
The purpose of this article is to calculate the numerical values of the Shannon (S) and Fisher (I) information entropies along with their Shannon power (J) and Fisher-Shannon product (P) for a free particle in three dimensions. In both coordinate and momentum space, the numerical values have been calculated for various values of the cubical box's length (L). Further, the validity of the entropic uncertainty relations such as the Bialynicki-Birula and Myceilski (BBM) inequality relation, Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and the Fisher-Shannon product relation have been investigated. The Fisher-Shannon product has been introduced and analyzed, which can be considered as a new correlation measure. The article is basically based on the concept, measurements and on the behaviour of the Shannon entropy, Fisher information entropy, Shannon power and the Fisher-Shannon product. Here, the plane wave solution of the free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The Fourier transforms of the coordinate space wave function provided the momentum space wave function. The probability densities corresponding to the respective wave functions have been employed to compute the numerical values of the information theoretic quantities such as Shannon entropy, Fisher entropy, Shannon power and the Fisher-Shannon product both in the coordinate and momentum space and the computed values have been presented in the respective tables. Finally, the entropic uncertainty relations investigated have been found consistently satisfying the corresponding inequalities such as Bialynicki-Birula and Mycielski (BBM) inequality, the Stam-Cramer-Rao inequalities and the Fisher-Shannon product relation. The study establishes the validity of the entropic uncertainty relations for the motion of a free particle in three dimensions.
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.
In this study, the energy equation and normalized wave function were obtained by solving the Schrödinger equation analytically utilizing the Eckart-Hellmann potential and the Nikiforov-Uvarov method. Fisher information and Shannon entropy were investigated. Our results showed higher-order characteristic behavior for position and momentum space. Our numerical results showed an increase in the accuracy of the location of the predicted particles occurring in the position space. Also, our results show that the sum of the position and momentum entropies satisfies the lower-bound Berkner, Bialynicki-Birula, and Mycieslki inequality and Fisher information was also satisfied for the different eigenstates. This study's findings have applications in quantum chemistry, atomic and molecular physics, and quantum physics.