古典電子在重核場下的軌跡分析

Abstract

本篇論文探討三體運動的軌跡，而其軌跡是相當的複雜多變，只要初始條 件一小小變化，就極有可能產生一條完完全全不同型態的軌跡，所以本算 論文用各種方法去對此複雜的軌跡作個歸納與分析。第一章首先簡單介紹 一輕粒子在中心力場運動的軌跡型態，這些軌跡可由運動方程式所導出的 錐線方程式 來作分類，但對於一輕粒子在兩固定中心力場下運動的軌跡 卻無法得到一個解析解來加以分類，而只能用電腦數值分析方法來獲得軌 跡，然後再作進一步探討。而第二章是利用古典電子在特殊的初始條件下 其軌跡有著規律性的變化，並由電子的Hamiltonian 討論著電子能量與兩 重核距離之關係，到最後用更系統的方法（變動能量和兩核距離）討論出 在什麼情況之下可得到哪種軌跡，並引用相關論文上有關固定兩重核下的 一些運動常數公式，而作更有意義的解釋。第三章，假定兩核移動，探討 電子的能量變化情形，而剛好在兩核接近到某距離時其電子變為最低的能 量，此現象與H2+分子在某距離穩定結合時的總能量為最低有著類似對應 ，但做出的結果份與實際物理情況不怎麼吻合，故接下來便討論如何使整 個理論模型更加嚴謹完備。在晸後的第四章裡探討者電子在多個重核底下 的軌跡現象，發現一個結果：在適當的初始條件下，光只有兩核A和B時由 A跑至B所花的時間比在兩核A、B之間還有其他重核存在時還多，因此我們 可以利用多重核之下所建立庫侖場，為電子找到一條從甲地跑到乙地所花 的時間為最少路徑。本文使用MATHEMATICA做為電腦模擬工具。 We discuss the trajectories of 3-bodies problems in this thesis. The trajectories are quite complicated. There are totally difference type oftrajectories, as long as a very small change made in initial conditions. Asa result, we use many methods to analyze the complicated trajectories. We briefly review the trajectories of electron in central force field inchpter 1. These trajectories can be classified by conic equation. But there is no analytic solution for an electron moving around two attractive centers. Weshould use numerical method to get the trajectories. In chapter 2, we find that the trajectories will change regularly underspecial initial conditions. Wediscuss the relationships between the energiesof electrons and two heavy nuclei. At last, we use more systematic method to discuss the trajectories. We also try to explain the results. We discuss the change of energies under 2 moving nuclei in chapter 3.When 2 nuclei approach to some distance , the energies of electron2 will bethe lowest. This corresponds bonding of H2+ . But the calculations didn'tmatch to physical situations quite well. So we discuss how to make the modelmore realistic in the next section. In the last chapter, we study the trajectories under the forces of multi-nuclei. We found that if there are only two nuclei A and B , the time for anelectron to travel from A to B is more than it spends when there is anotherheavy nucleus between A and B . Under suitable initial conditions, we candesign a shortest-time path connecting two given locations. The computer simulation tool used in this thesis is Mathematica[1].