The solutions of two potentials with one potential made up of a combination of constant, Yukawa, and inversely quadratic potentials and the other made up of constant, Coulomb, and inversely quadratic potentials are obtained under the radial Schrödinger equation using the elegant parametric Nikiforov–Uvarov method. The energy equations and their corresponding wave functions are obtained in a close and compact form. The Fisher information for configuration space and momentum space are obtained for each combination of the potentials. It has been revealed that the energy eigenvalues of each combined potential model has a turning point. It is also shown that one special case in one combined potentials and another special case in the other combined potentials have equivalent energy eigenvalues. The results for the constant potential as a subset potential in each combination are not exactly the same. The Fisher information for each combined potentials and their respective subset potentials satisfied Fisher information-based uncertainty relation. It is also shown that the effect of the screening parameter on the Fisher information at the ground state and at the first excited state for one of the combining potential has a diffused format.
The solutions of two potentials with one potential made up of a combination of constant, Yukawa, and inversely quadratic potentials and the other made up of constant, Coulomb, and inversely quadratic potentials are obtained under the radial Schrödinger equation using the elegant parametric Nikiforov–Uvarov method. The energy equations and their corresponding wave functions are obtained in a close and compact form. The Fisher information for configuration space and momentum space are obtained for each combination of the potentials. It has been revealed that the energy eigenvalues of each combined potential model has a turning point. It is also shown that one special case in one combined potentials and another special case in the other combined potentials have equivalent energy eigenvalues. The results for the constant potential as a subset potential in each combination are not exactly the same. The Fisher information for each combined potentials and their respective subset potentials satisfied Fisher information-based uncertainty relation. It is also shown that the effect of the screening parameter on the Fisher information at the ground state and at the first excited state for one of the combining potential has a diffused format.
Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.
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.
The approximate analytical solutions of the radial Schrӧdinger equation have been obtained with a newly proposed potential called Hellmann-generalized Morse potential. The potential is a superposition of Hellmann potential and generalized Morse or Deng-Fan potential. The Hellmann-generalized Morse potential actually comprises of three different potentials which includes Yukawa potential, Coulomb potential and Deng-Fan potential. The aim of combining these potentials is to have a wide application. The energy eigenvalue and the corresponding wave function are calculated in a closed and compact form using the parametric Nikiforov-Uvarov method. The energy equation for some potentials such as Deng-Fan, Rosen Morse, Morse, Hellmann, Yukawa and Coulomb potentials have also been obtained by varying some potential parameters. Some numerical results have been computed. We have plotted the behavior of the energy eigenvalues with different potential parameters and also reported on the numerical result. Finally, we computed the variance and information energy for the Hellmann-generalized Morse potential.
The radial Schrödinger equation was solved with the combination of three important potentials with [Formula: see text] as deformed parameter via the parametric Nikiforov–Uvarov method and the energy equation as well as the corresponding normalized radial wave function were obtained in close and compact form. The energy equation obtained was used to study eight molecules. The effect of the deformed parameter on energy eigenvalues was also studied numerically. The subset of the combined potential was also studied numerically and the results were found to be in agreement with the previous results. To extend the application of our work, the wave function obtained was used to calculate some theoretic quantities such as the Tsallis entropy, Rényi entropy and information energy. By putting the Tsallis index to 2, we deduced the information energy from Tsallis entropy. Finally, the effect of the deformed parameter and screening parameter, respectively, on the theoretic quantities were also studied.
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.
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.