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International Journal of Quantum Chemistry

Quantum information of the modified Mobius squared plus Eckart potential

Ifeanyi Jude Njoku, Emeka Onyeocha, Chibueze Paul Onyenegecha, Modestus Onuoha, Eugene K. Egeonu, Placid Nwaokafor

Published Nov 21, 2022• 12 citations
Publisher

Abstract

Abstract The quantum information measures and complexity of the modified Mobius squared plus Eckart (MMSE) potential are presented in this paper. First, the energy eigenvalues and wave function of the system are obtained from the approximate solutions of the Schrödinger equation via the parametric Nikiforov‐Uvarov (pNU) method. Using the wave function, the Shannon entropy, Onicescu information energy and Fisher information of the system are examined for two low‐lying states along with the modified Lopez‐Ruiz‐Mancini‐Calbet (LMC) complexity and Heisenberg uncertainty relation. The results of the work point to the fact that the radial (momentum) probability density peak shifts to lower (higher) values with increase in the screening parameter. Furthermore, the Bialynicki‐Birula and Mycielski (BBM) inequality, the lower bound of the modified LMC complexity, the Fisher information sum inequality and the Stam‐Cramer‐Rao inequality are verified for the system. Also, the Heisenberg uncertainty principle is verified for the MMSE potential and the existence of squeezed states is observed for both position and momentum states.

DOI
10.1002/qua.27050

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Abstract The quantum information measures and complexity of the modified Mobius squared plus Eckart (MMSE) potential are presented in this paper. First, the energy eigenvalues and wave function of the system are obtained from the approximate solutions of the Schrödinger equation via the parametric Nikiforov‐Uvarov (pNU) method. Using the wave function, the Shannon entropy, Onicescu information energy and Fisher information of the system are examined for two low‐lying states along with the modified Lopez‐Ruiz‐Mancini‐Calbet (LMC) complexity and Heisenberg uncertainty relation. The results of the work point to the fact that the radial (momentum) probability density peak shifts to lower (higher) values with increase in the screening parameter. Furthermore, the Bialynicki‐Birula and Mycielski (BBM) inequality, the lower bound of the modified LMC complexity, the Fisher information sum inequality and the Stam‐Cramer‐Rao inequality are verified for the system. Also, the Heisenberg uncertainty principle is verified for the MMSE potential and the existence of squeezed states is observed for both position and momentum states.
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.
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.
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.
Abstract In this article, some information theoretic concepts are analyzed for modified Hylleraas plus exponential Rosen Morse potential in position and momentum space. The angular and radial contributions of the information density are graphically demonstrated for different states. The entropy densities have asymmetric shape which depends on the values of quantum numbers. The information entropy is analytically obtained for ground state of the potential whereas the numerical calculations have been performed for the higher states and Bialynicki‐Birula and Mycielski inequality is tested for various states using different parameters of the potential. It is shown that the information entropy is reduced, both in position and momentum space, for careful selection of some parameters. Further, it is found that there exist eigenstates exhibiting squeezing in information entropy of modified Hylleraas plus exponential Rosen Morse and Eckart potential. Interestingly, in case of Eckart potential, the squeezed states are obtained in position as well as momentum space and are attempted to saturate for some values of the parameters.
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.
We solved the Schrödinger equation with a certain approximation to the centrifugal term for an arbitrary angular momentum state with the Eckart Manning Rosen potential. The bound-state energy eigenvalues and the corresponding wave functions have been approximately obtained using the parametric Nikiforov Uvarov method. The solutions of the Schrödinger equation for the Eckart potential, Manning Rosen potential, and Hulthén potential have been obtained using a certain transformation. The concepts of the Shannon entropy and the Fisher information of a system under the Eckart Manning Rosen potential are investigated in detail. The behavior of the screening parameter and the quantum number n for Fisher information and the Shannon entropy are also investigated.