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International Journal of Quantum Chemistry

Shannon entropy and Fisher information for screened Kratzer potential

P. O. Amadi, Akpan N. Ikot, Alalibo Thompson Ngiangia, U. S. Okorie, Gaotsiwe Joel Rampho, Hewa Yaseen Abdullah

Published Apr 25, 2020• 49 citations
Publisher

Abstract

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.

DOI
10.1002/qua.26246

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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.
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.
In this research, the radial Schrödinger equation for a newly proposed screened Kratzer–Hellmann potential model was studied via the conventional Nikiforov–Uvarov method. The approximate bound state solution of the Schrödinger equation was obtained using the Greene–Aldrich approximation in addition to the normalized eigenfunction for the new potential model, both analytically and numerically. These results were employed to evaluate the rotational–vibrational partition function and other thermodynamic properties for the screened Kratzer–Hellmann potential. The results obtained have been graphically discussed. Also, the normalized eigenfunction has been used to calculate some information-theoretic measures including Shannon entropy and Fisher information for low-lying states in both position and momentum spaces numerically. The Shannon entropy results obtained agreed with the Bialynicki-Birula and Mycielski inequality, while the Fisher information results obtained agreed with the Stam, Cramér–Rao inequality. Also, alternating increasing and decreasing localization across the screening parameter for both eigenstates was observed.
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.
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.
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.
In this article, the plane wave solution for a free particle in three dimensions is considered and the wave function is normalized in an arbitrarily large but finite cube. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities are employed to compute the numerical values of the information theoretic quantities such as Shannon information entropy (S), Fisher information entropy (I), Shannon power (J) and the Fisher–Shannon product (P) both in coordinate and momentum spaces for different values of the length (L) of the cubical box. Numerical values so found satisfy the Beckner, Bialynicki-Birula and Myceilski (BBM) inequality relation; Stam-Cramer-Rao inequalities (better known as the Fisher based uncertainty relation) and Fisher-Shannon product relation. This establishes the validity of the information theoretic inequalities in respect of the motion of a free particle.