Abstract
In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigensolutions and total normalized wave function of Schrödinger equation expressed in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic Potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential). The proposed potential is best suitable for smaller values of the screening parameter α . The resulting energy eigenvalue is presented in a close form and extended to study thermal properties and superstatistics expressed in terms of partition function Z and other thermodynamic properties such as vibrational mean energy U , vibrational specific heat capacity C , vibrational entropy S , and vibrational free energy F . Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter ( α ) as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases.
Highlights
The approximate analytical solutions of one-dimensional radial Schrödinger equation with a multiple potential function have been studied using a suitable approximation scheme to the centrifugal term within the frame work of the parametric Nikiforov-Uvarov method [1]
Eigensolutions for both relativistic and nonrelativistic wave equations have been studied with different methods which include the following: Exact quantisation, WKB, NikiforovUvarov method (NU), Laplace transform technique, Advances in High Energy Physics asymptotic iteration method, proper quantisation, supersymmetric quantum mechanics approach, vibrational approach, formula method, factorisation method, and Shifted 1/N-expansion method [8–13]
The practical application of energy eigenvalue of Schrödinger equation in investigating the partition function, thermodynamic properties, and superstatistics arouses the interest of many researchers
Summary
The approximate analytical solutions of one-dimensional radial Schrödinger equation with a multiple potential function have been studied using a suitable approximation scheme to the centrifugal term within the frame work of the parametric Nikiforov-Uvarov method [1]. Okon et al [54] obtained the thermodynamic properties and bound state solutions of the Schrödinger equation using Mobius square plus screened Kratzer potential for two diatomic systems (carbon(II) oxide and scandium fluoride) within the framework of the Nikiforov-Uvarov method. Their results were in agreement to semiclassical WKB among others. Oyewumi et al [56] studied the thermodynamic properties and the approximate solutions of the Schrödinger equation with shifted Deng-Fan potential model within the framework of asymptotic Iteration method where they apply Pekeris-type approximation to centrifugal term to obtain rotational-vibrational energy eigenvalues for selected diatomic systems. Boumali and Hassanabadi [61] studied thermal properties of a twodimensional Dirac oscillator under an external magnetic field where they obtained relativistic spin-1\2 fermions subject to Dirac oscillator coupling and a constant magnetic field in both commutative and noncommutative spaces
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