Approximate Solutions, Thermal Properties and Superstatistics Solutions to Schr\"odinger Equation

TL;DR

This paper derives analytical solutions for the Schrödinger equation with a Coulomb plus Screened Exponential Hyperbolic potential, explores thermal and superstatistics properties, and validates results with numerical simulations and existing literature.

Contribution

It introduces a new potential model and provides analytical eigen solutions, thermal properties, and superstatistics analysis, extending to special cases like Hellmann and Yukawa potentials.

Findings

Eigen solutions expressed in terms of Jacobi polynomials.

Thermal properties and superstatistics show excellent agreement with literature.

Numerical solutions obtained using MATLAB and Mathematica.

Abstract

In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigen solutions and total normalized wave function of Schr\"odinger equation express 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 .The resulting energy eigen equation is presented in a close form and extended to study thermal properties and superstatistics express 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 ( alpha) 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.