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系統識別號 U0026-0812200915223419
論文名稱(中文) 時空守恆法模擬薛丁格方程式
論文名稱(英文) Simulation of the Schrödinger Equation by the Space-Time Conservation Element and Solution Element Method
校院名稱 成功大學
系所名稱(中) 工程科學系碩博士班
系所名稱(英) Department of Engineering Science
學年度 97
學期 2
出版年 98
研究生(中文) 王甫宇
研究生(英文) Fu-Yu Wang
電子信箱 subow1009@gmail.com
學號 n9696141
學位類別 碩士
語文別 中文
論文頁數 75頁
口試委員 口試委員-曾子彝
口試委員-駱文傑
指導教授-楊瑞珍
口試委員-黃吉川
口試委員-傅龍明
中文關鍵字 薛丁格方程式  勢能井  時空守恆法 
英文關鍵字 CESE method  Schrödinger equation  Potential Barrier 
學科別分類
中文摘要 本論文主要研究目的為利用時空守恆法(CESE method)模擬線性與含勢能項之線性薛丁格方程式,其中以線性薛丁格方程式探討自由粒子之行為,和含勢能項之線性薛丁格方程式探討波函數於位能井內之分佈。薛丁格方程式主要是描述粒子的機率分佈,所謂的自由粒子是指粒子在運動過程中不受外力的影響,而含有勢能項的薛丁格方程式,是在探討當粒子碰撞勢能井邊界後所產生的機率分佈情形,其中我們依量子力學的觀點可以知道,在與勢能井碰撞的過程中,粒子是存在穿透與反射兩種機率,而主要影響因素為粒子的總能(E)和位能井(V)的大小,當E>V時,粒子存在穿透與反射,而當E V時,粒子會近似於全反射。
時空守恆法(CESE method)以物理守恆的觀點出發,利用統御方程式滿足通量守恆進行離散,為時間和空間上均具有二階準確度的嶄新數值方法。以時空守恆法數值解與自由粒子之解析解比較可知,時空守恆法模擬線性的薛丁格方程式的機率分佈時,仍具有相當精確之結果;在模擬含有勢能項的線性薛丁格方程式中,當粒子與勢能井碰撞後仍可以得到合理的數值解。此外,研究中還以von Neumann穩定性分析探討時空守恆法在解含勢能項之線性薛丁格方程式時的穩定性。
英文摘要 The main purpose of this research is to utilize the space-time Conservation Element and Solution Element (CESE) method to simulate the linear Schrödinger equation. Where the linear Schrödinger equation describes a free particle behavior, and the Schrödinger equation models the distribution of the wave function within a potential barrier. Schrödinger equation mainly explains the probability distribution of the particle. A free particle motion obeys the Schrödinger equation without external force. Moreover, the Schrödinger equation with potential function can calculate the probability distribution after the wave function collides with the potential barrier. In view of quantum mechanics, we know that in the course of colliding with potential barrier, the particle presents tunneled probability and reflected probability. The main influence factors are the total energy (E) of the particle and the potential barrier (V). When E>V, the particle presents tunnel and reflection; when E V, the particle exhibits completely reflection.
CESE method follows the principle of physical conservation, and derives the discrete formulation of governing equation by means of flux conservation. In time and space, the CESE method is a new numerical scheme with second order accuracy. Comparing the CESE results with the exact solution on the free particle behavior, we demonstrate know that the CESE method can still maintain an accurate result even when the simulation of the probability distribution has a dramatic change. In the simulation of the Schrödinger equation within the potential barrier, the CESE method still received reasonable results after the collision finished. Furthermore, by using the von Neumann stability analysis, this study presents the restriction of numerical parameters of the CESE method for solving the Schrödinger equation.
論文目次 中文摘要 I
Abstract II
誌謝 IV
目錄 V
表目錄 VII
圖目錄 VIII
符號表 X
第一章 導論 1
1-1研究動機 1
1-2 文獻回顧 2
1-3 本文架構 3
第二章 一維 CESE 之介紹與應用 4
2-1 序論 4
2-2 推導和驗證(線性方程式) 5
2-2.1 線性方程式( ) 5
2-2.2 利用純量函數的推導 11
2-2.3 程式驗證 14
2-3 推導和驗證(非線性方程式) 18
2-3.1非線性方程式(a-α Scheme) 18
2-3.2 程式驗證 21
第三章 薛丁格方程式 25
3-1序論 25
3-2薛丁格方程式 25
3-3線性薛丁格方程 28
3-4含勢能項之線性薛丁格方程 29
第四章 薛丁格方程式之模擬與討論 31
4-1序論 31
4-2 自由粒子 31
4-3 含勢能項之線性薛丁格方程式 41
4-3.1台階位能(Step potential) 42
4-3.2障壁位能(Barrier potential) 44
4-3.3方井位能(Square-well potential) 51
4-4 穩定性分析 58
第五章 結論與建議 60
5-1 結論 60
5-2建議和未來展望 61
參考文獻 62
附錄 A 64
附錄 B 65
附錄 C 66
自述 75
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