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博碩士論文 etd-0423116-131419 詳細資訊
Title page for etd-0423116-131419
論文名稱
Title
使用路丁對角化法計算狄拉克之低能量等效哈密頓
Calculations of the Effective Dirac Hamiltonian in the Low-energy Limit by Using Löwdin Partitioning Method
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
45
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2016-01-19
繳交日期
Date of Submission
2016-05-24
關鍵字
Keywords
路丁對角化方法、狄拉克方程、T-BMT方程
Löwdin partitioning method, Dirac equation, T-BMT Equation
統計
Statistics
本論文已被瀏覽 5776 次,被下載 701
The thesis/dissertation has been browsed 5776 times, has been downloaded 701 times.
中文摘要
低能量的狄拉克方程( g=2與g≠2 )之哈密頓與T-BMT的古典對應之嚴格數學證明已由Chen與Chiou在[1,2,3]給出。其所證明之方式為:使用Foldy-Wouthuysen(福迪-沃希)與Kuzelnigg之微擾展開法並利用數學歸納法完成證明。本篇論文是使用路丁(Löwdin)微擾方法展開。文獻中,Chen 與Chiou 使用Foldy-Wouthuysen方法展開到ε^5 [1],其中ε=1/2m0c^2 ,而使用Kuzelnigg方法展開到ε^11 [3]。而此篇論文僅計算g=2 的情況,計算至ε^5 。在每一階的狄拉克哈密頓都有非對角化的部分,我們利用路丁的方法找到某個非對角化係數的解去消除原本狄拉克哈密頓的非對角化部分,接著計算出每一階符合非對角化消去條件的對應係數。由此我們就能得到哈密頓對角化的部分,而非對角化的部分則出現在更高階。必須強調的是我們僅計算均勻且線性電磁場的結果。此外,路丁對角化方法與福迪-沃希森轉換這兩種方法,直至ε^5 的結果是完全相同的。最後,我把路丁對角化方法的結果跟T-BMT方程做比較,並再次得到在低能量近似下的狄拉克哈密頓和T-BMT哈密頓是一樣的。
Abstract
The proof of classical correspondence of Dirac equation( g=2and g≠2 ) in low energy and T-BMT equation was given by Chen and Chiou[1, 2, 3]. They use Foldy-Wouthuysen and Kuzelnigg perturbative expansion. Further, they take advantage of Mathematical induction and make testimony complete. In[1], Chen and Chiouuse Foldy-Wouthuysen method to expand to ε^5 where ε=1/2m0c^2 , and use Kuzelnigg method to expand to ε^11 [3]. In my thesis, I merely expand to ε^5 by Löwdin partitioning method at g=2 . In this thesis, I will present the detailed calculation procedure of the Löwdin partitioning method. The key part of this method is to find some non-diagonal forms to delete the original non-diagonal forms of Dirac Hamiltonian and to calculate the every order coefficients of Dirac Hamiltonian, but the parts of off-diagonal terms influence the higher orders. It is emphasized that the results which we calculated is only in uniform and linear electromagnetic field. The two traditional method of diagonalizing Hamiltonian, Löwdin partitioning method and Foldy-Wouthuysen Transformation, lead to the same results in ε^5 . Finally, I will compare with the result of Löwdin partitioning method and T-BMT equation, and agree that the Dirac Hamiltonian in low-energy limits the same with T-BMT Hamiltonian again.
目次 Table of Contents
論文審定書 i
序言 ii
摘要 iii
Abstract iv
Contents v
第一章 狄拉克方程的物理意義 1
1.1. 自旋物理簡介 1
1.2. 量子力學中的狄拉克方程 2
1.3. T-BMT方程 5
第二章 理論方法:微擾近似 8
2.1. 福迪-沃希轉換 8
2.2. 路丁對角化方法 17
2.3. 第五階的計算 22
第三章 結果與討論 31
Bibliography 36
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