Approximate Solutions, Thermal Properties, and Superstatistics Solutions to Schrödinger Equation

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.

Exact solutions of $$\kappa$$-dependent Schrödinger equation with quantum pseudo-harmonic oscillator and its applications for the thermodynamic properties in normal and superstatistics

“…Superstatistics is known to be a superposition of different statistical models in statistical physics, which helps in studying non-linear and non-equilibrium systems 23 , 24 . By taking a Laplace’s transform of the probability density function within the concept of superstatistics, one can obtain the generalized Boltzmann factor defined as 31 , 32 …”

“…The probability density function obeys the normalization condition . Thus, the effective Boltzmann factor for the modified delta distribution function is given as 31 , 32 , …”

“…It can be seen in Eq. ( 14 ) that when , the superstatistics reduced to ordinary statistic mechanics 31 , 32 . Within the superstatistics, the thermodynamics functions of the system are valid for all values of q and all the thermodynamic functions as well as the energy depend on the system.…”

“…Superstatistics is known to be a superposition of different statistical models in statistical physics, which helps in studying non-linear and non-equilibrium systems 23,24 . By taking a Laplace's transform of the probability density function f β ′ , β within the concept of superstatistics, one can obtain the generalized Boltzmann factor defined as 31,32 The probability density function obeys the normalization condition f β ′ , β = δ β ′ − β . Thus, the effective Boltzmann factor for the modified delta distribution function is given as 31,32 , Here, q is the deformation parameter which lies between 0 ≤ q ≤ 1 , E is the energy level of the system.…”

“…In this formalism, two different statistics were superimposed to explain non-equilibrium systems [25][26][27] . Much study have been carried out with superstatistics within equilibrium and non-equilibrium statistical mechanics framework [28][29][30][31][32] . The effects of cosmic-string parameters on harmonic oscillator have been considered using both ordinary statistics and superstatistics formalism 33 .…”