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Scientific Reports

Relativistic bound state solutions and quantum information theory in D dimensions under exponential-type plus Yukawa potentials

R. Horchani, E. Omugbe, Ifeanyi Jude Njoku, Laura Milena Pérez, C. A. Onate, Arezu Jahanshir, Elmustapha Feddi, K. O. Emeje, Edwin Samson Eyube

Published Nov 19, 2024• 7 citations
Publisher

Abstract

The bound-state solution of the radial Klein-Gordon equation has been obtained under the interaction of an exponential-type and Yukawa potential functions. The Greene-Aldrich approximation has been used to overcome the centrifugal barrier and enable the analytical solutions of the energy and wave functions in closed form. The momentum space wave function in D dimensions has been constructed using the Fourier transform. The mean values have been conjectured for the position and momentum spaces using two equivalent equations. The effects of the potential parameters on the expectation values and quantum information measurement have been investigated. For the 1D case, the results obey the Heisenberg uncertainty principle, Fisher, Shannon, Onicescu, and the Rényi entropic inequalities. Other information complexities measures, such as Shannon Power, Fisher-Shannon, and Lopez-Ruiz-Mancini-Calbet, have been verified. For the ground state, the 1D momentum expectation value [Formula: see text] coincides with the 3D [Formula: see text] values, which is an indication of degeneracy. The total energy of a particle in both 1D and 3D space may be degenerate due to the inter-dimensional degeneracy of the quantum numbers. However, in this present result, the degeneracy in 1D and 3D occurred for fixed quantum states at different momentum intervals. Thus, in 1D, a particle may transit an entire space ([Formula: see text] with a certain kinetic energy, which must be equal to its kinetic energy if it moves through the interval [Formula: see text] in 3D space. This may have implications for kinetic energy degeneracy in higher dimensions.

DOI
10.1038/s41598-024-80123-9

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The bound-state solution of the radial Klein-Gordon equation has been obtained under the interaction of an exponential-type and Yukawa potential functions. The Greene-Aldrich approximation has been used to overcome the centrifugal barrier and enable the analytical solutions of the energy and wave functions in closed form. The momentum space wave function in D dimensions has been constructed using the Fourier transform. The mean values have been conjectured for the position and momentum spaces using two equivalent equations. The effects of the potential parameters on the expectation values and quantum information measurement have been investigated. For the 1D case, the results obey the Heisenberg uncertainty principle, Fisher, Shannon, Onicescu, and the Rényi entropic inequalities. Other information complexities measures, such as Shannon Power, Fisher-Shannon, and Lopez-Ruiz-Mancini-Calbet, have been verified. For the ground state, the 1D momentum expectation value [Formula: see text] coincides with the 3D [Formula: see text] values, which is an indication of degeneracy. The total energy of a particle in both 1D and 3D space may be degenerate due to the inter-dimensional degeneracy of the quantum numbers. However, in this present result, the degeneracy in 1D and 3D occurred for fixed quantum states at different momentum intervals. Thus, in 1D, a particle may transit an entire space ([Formula: see text] with a certain kinetic energy, which must be equal to its kinetic energy if it moves through the interval [Formula: see text] in 3D space. This may have implications for kinetic energy degeneracy in higher dimensions.
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.
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.
The charge spreading of the ground and excited states of Klein-Gordon particles moving in a Coulomb potential is quantitatively analysed by means of ordinary moments and the Heisenberg measure as well as by using the most relevant information-theoretic measures of global (Shannon entropic power) and local (Fisher information) types.The dependence of these complementary quantities on the nuclear charge Z and the quantum numbers characterizing the physical states is carefully discussed.The comparison of relativistic Klein-Gordon and non-relativistic Schrödinger values is made.Nonrelativistic limits at large principal quantum number n and for small values of Z are also reached.Contents 1. Introduction 2 2. Radial expectation values and Heisenberg's measure 5 3. Shannon and Fisher information measures 8 4. Conclusions 12
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 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.
This work present analytical results for the position- and momentum-space information entropies, of the 1s2-state of helium-like ions, using different interaction potentials. The potentials that we used are the Yukawa potential (YP), and the exponential-cosine-screened Coulomb potential (ECSCP). The investigated studies allow us to relate the position-space information with the momentum-space information of Shannon and Fisher, as well as Shannon entropy power, and the Fisher-Shannon information product, through different famous relations. The calculation is done using one-electron charge density of entangled two-parameter wave function. On one hand, the results that are presented for fifteen members in the helium iso-electronic sequence demonstrate with precision the effect of correlation on bare charge distributions. On the other hand, it leads to some very important results for both the correlated and uncorrelated values of the informatic entropies. Analytical formula for the momentum-space information entropies are given. The effect of the nuclear charge and the screening parameter on the informatic expressions has been studied for both potentials. Detailed computational and numerical values and characteristics of these quantities are reported here for the first time. New inequality has been proposed with Fisher’s total value to measure the correlation of two electrons.