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Eurasian Physical Technical Journal

COMPUTATIONAL ANALYSIS OF INFORMATION-THEORETIC MEASURES AND OSCILLATOR STRENGTHS IN QUANTUM SYSTEMS VIA THE NIKIFOROV–UVAROV METHOD

Etido Patrick Inyang, P.O. Okoi, I. M. Nwachukwu

Published Dec 29, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.31489/2025n4/101-116

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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.
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Quantum information entropies have been properly studied within the confines of nonrelativistic quantum mechanics ( QM ) by solving Schrödinger wave equation for different considerable potential models to obtained the energy and total normalised wave function which are essential tools for determining various information entropies. However, not much research has been done on evaluation of information entropies within the relativistic quantum mechanics because of its complexities and existence of both multiple and quadratic eigenvalues for fixed quantum state ( n ) and orbital angular quantum number (ℓ)(\ell )(ℓ) . In this work, we investigate Fisher information and Shannon entropy in three dimensional space within the relativistic regime by solving Klein-Gordon ( KG ) equation for equal scalar and vector potential with Schioberg plus Yukawa potential ( SPYP ) using the Parametric Nikiforov-Uvarov ( PNU ) method and compare the results to its nonrelativistic counterpart. It is important to note that the numerical solutions for the three dimensional Fisher information and Shannon entropy obtained for both the relativistic and nonrelativistic wave equations using SPYP adequately satisfies Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality. The radial wave function, radial probability density, momentum space wave function and momentum probability density graphical plots were all obtained for both cases respectively. The numerical computations for Fisher information and Shannon entropy in three dimensional space were carried out for ℓ=0\ell = 0ℓ=0 with different quantum states n=0n= 0n=0 and 1. The investigation of Fisher information and Shannon entropy are relevance in describing localisation and delocalisation mechanism of particles in atomic orbitals within a potential well, relativistic confinement effect and entropy-energy correlation.
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