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標題: | 變分分離動能法求解少體系統 Variational Kinetic Energy Partition Method for Few-body Systems |
作者: | Yuh-Shiuan Liu 劉育軒 |
指導教授: | 趙聖德(Sheng-Der Chao) |
關鍵字: | 薛丁格方程式,分離動能法,負質量,多體系統,變分法,氦原子,氫分子,四中心積分問題, schrodinger equation,kinetic energy partition,negative mass,few-body systems,variational principle,helium atom,hydrogen molecule,four-center integrals, |
出版年 : | 2020 |
學位: | 碩士 |
摘要: | 本文延續本實驗室發展的一個新穎且具有物理意義和系統性的方法―分離動能法(Kinetic Energy Partition),其原理為將漢米爾頓系統中的動能項的質量拆解,拆解後的各個動能項分別與對應的位能項組成各個子系統,而子系統的總和仍為原來的漢米爾頓系統。我們會先介紹KEP方法的理論,以最簡單的單體系統開始,延伸至少體系統,其中在少體系統中,引入「負質量」的概念來處理斥力位能問題。最後使用Hartree-like形式的試探波函數,來求得能量。接著為了減少分離動能法的計算難度,我們引入絕熱近似法(Adiabatic Approximation),稱其為AKEP。不過在計算了KEP與AKEP能量後,發現仍與正合解有一定的誤差,為了改善此結果,我們引入變分法(Variational Principle),將波函數中的質量參數視為變數進行變分,得出的最佳化能量即為VKEP。本文將會以不同模型範例作以及真實粒子―氦原子作為例子,以用來驗證VKEP的可行性。
此外另一個部分為真實分子―氫分子,為探討分離動能法在分子系統上的可行性,我們遇到了積分技術上的問題―四中心積分問題,在四中心積分問題上,我們利用擬合的方式,擬合出其替代函數,成功解決部分積分無法處理的問題。 This study continues a new, physical and systematic method, kinetic energy partition, developed by our group for solving schrodinger equations. The critical point of the KEP method is to separate the mass factor of the kinetic energy terms into effective ones, each to be associated with partial terms, so the whole Hamiltonian can be written as the sum of the partial subsystem. We will introduce the theory of the KEP method first, starting from the simple one-particle systems, and extending to few-body systems. For few-body systems, we propose to use the concept of “negative mass” to deal with the repulsive potential. Finally using the trial wave function of the Hartree-like forms to get the energy of the KEP method. we propose to use the “adiabatic approximation” to reduce the calculating time of the KEP method, called “AKEP”. But after calculating KEP and AKEP, we find that there is still a error of the KEP method. To prove the result, we propose to use the variational principle to variate the mass factor of the wave function, called “VKEP”. We will discuss different models including the particle-helium atom. Another part is the hydrogen molecule, to test the KEP method in molecule system, we calculate and solve the problem – four center integrals. In four center integrals, we use the method of fitting to get the alternative function. Successfully solved the problem which can not be handled. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/66867 |
DOI: | 10.6342/NTU201904444 |
全文授權: | 有償授權 |
顯示於系所單位: | 應用力學研究所 |
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