薛定谔方程的近似解、热性质和超统计解
Approximate Solutions, Thermal Properties and Superstatistics Solutions to Schrödinger Equation
摘要：
本文利用参数Nikiforov-Uvarov方法，利用库仑加屏蔽指数双曲势（CPSEHP）得到了用Jacobi多项式表示的薛定谔方程的本征解和归一化波函数，得到了不同轨道角量子数下的概率密度图，以及一些特殊情况（Hellmann和Yukawa势）。所提出的势最适合于筛选参数的较小值。由此得到的能量本征方程以封闭形式给出，并扩展到研究热性质和超统计量，配分函数（Z）和其他热力学性质，如振动平均能量（U）、振动比热容（C）、振动熵（S）和振动自由能（F）。利用得到的能量方程，借助Matlab软件，通过Hellmann-Feynman定理（HFT）得到了不同屏蔽参数（alpha）值以及不同期望值下的束缚态数值解.计算得到的配分函数和其他热力学性质的变化趋势与已有文献的结果吻合较好。由于分析数学的复杂性，使用Mathematica 10.0版本软件评估超统计和热性质。所提出的势模型作为特例可简化为Hellmann势、Yukawa势、Screened Hyperbolic势和Coulomb势。
英文摘要：
In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigen solutions and total normalized wave function of Schrödinger 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.
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