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系統識別號 U0026-0812200915204975
論文名稱(中文) 由複數量子軌跡觀點討論量子混沌與量子機率論
論文名稱(英文) A Study on Quantum Chaos and Quantum Probability from the Viewpoint of Complex Quantum Trajectories
校院名稱 成功大學
系所名稱(中) 航空太空工程學系碩博士班
系所名稱(英) Department of Aeronautics & Astronautics
學年度 97
學期 2
出版年 98
研究生(中文) 魏嘉宏
研究生(英文) Chia-Hung Wei
電子信箱 p4891121@mail.ncku.edu.tw
學號 p4891121
學位類別 博士
語文別 英文
論文頁數 175頁
口試委員 口試委員-黃榮興
口試委員-黃皇男
口試委員-容志輝
口試委員-張書銓
口試委員-孔健君
指導教授-楊憲東
召集委員-林三益
口試委員-李君謨
中文關鍵字 路徑積分軌跡  李氏指數  機率密度函數  多路徑現象  複數力學  量子力學  量子混沌  量子漢彌爾頓力學 
英文關鍵字 probability density function (PDF)  multi-path phenomenon  path integral trajectories  complex mechanics  quantum chaos  Lyapunov exponent  quantum Hamilton mechanics  Quantum mechanics 
學科別分類
中文摘要 本論文基於量子力學的複數軌跡詮釋法,從量子多路徑行為的討論開始,透過與費因曼路徑積分軌跡觀念上的聯結,說明引入虛部動態對多路徑現象的重要性。接著,多路徑行為的量子特性將被引用在量子混沌現象的討論上,傳統的認知中,一維、二維量子系統由於自由度上的限制,不能出現混沌現象,本論文藉著將動態變數拓展到複數空間的力學機制,獲取額外的自由度,結合量子系統的多路徑行為與額外的量子力,透過兩個簡化的物理系統:一維簡諧振子與二維帶電非等向性簡諧振子在磁場中的動態,證明量子混沌在低維度系統的可能性。要特別注意的是,本論文對量子混沌現象的討論過程中,由於粒子軌跡觀念的重新引入,使得量子力學對量子混沌的困惑得到解決。如同古典混沌的觀念,複數軌跡的發散性也能夠成為評估量子系統混沌現象的重要依據。本論文引入著名的混沌量化指標:李氏指數(Lyapunov exponent),來說明量子混沌的強弱特性,這也縮小了量子混沌與古典混沌的差異性。
最後,本論文創新嘗試將量子混沌現象與量子系統的機率性詮釋做結合,提出量子系統只能以統計性觀點來描述最主要的原因,可能來自於量子系統本身的混沌行為。雖然混沌現象使粒子軌道出現不規則且無法預測的複雜性,但是從長時間的統計觀點來看,混沌軌跡確實展現某些顯著的不變性。本論文將藉由二維帶電非等向性簡諧振子在磁場中的動態,來說明混沌性量子軌跡在空間中的分佈的確與由波函數所定義的機率分布有高度的相關性。由此也證明了,薛丁格方程式所解出的波函數,的確扮演著導引量子運動的關鍵角色。
英文摘要 On the basis of quantum Hamilton mechanics, several issues are addressed in this dissertation. First of all, we study the multi-path behavior of quantum systems by virtue of the complex trajectory interpretation of quantum mechanics. It is shown that Feynman’s path-integral trajectories can be represented by the complex trajectories and then parameterized within the framework of quantum Hamilton mechanics.
Next, two simplified physical systems, a 1D harmonic oscillator and a 2D charged anisotropic harmonic oscillator in a uniform magnetic field, are demonstrated to exhibit chaos from the viewpoint of particle-like behavior. While conventional quantum mechanics and Bohmian mechanics both predict that 1D harmonic oscillator shows no signature of chaotic behavior, we find that in quantum Hamilton mechanics this system exhibits both regular and chaotic behavior, depending on the composition of wavefunctions and on the particle’s initial position. We continue to investigate chaotic behavior in a 2D charged anisotropic harmonic oscillator. Even the possibility of chaos in eigenstates has been ruled out from Bohm’s trajectory interpretation, we still find obvious chaotic features in eigenstates of this 2D quantum oscillator. The territory of quantum chaos indeed can be enlarged via the complex-extended dynamics.
Finally, we point out that the complex chaotic dynamics may be the origin of the probability interpretation of quantum mechanics. In view of the generality of quantum chaos, it is impossible to predict the final states from the initial states for quantum systems. However, the statistical invariability of chaotic behavior offers another route for us to understand quantum systems. In this dissertation, the comparison between the distribution of complex quantum trajectories on the real coordinate space and the theoretic probability density function determined from the wavefunction shows that a chaotic quantum particle which seems to move irregularly is indeed guided by the wavefunction and attempts to appear somewhere with a statistical regularity. We also examine this tendency for Bohmian trajectories, however, no analogue of the complex trajectories can be found.
論文目次 ABSTRACT IN CHINESE ………………………i
ABSTRACT ……………………………..iii
CHINESE ABSTRACT OF EACH CHAPTER ………v
CONTENTS ………………….xiii
LIST OF FIGURES ………….xvi
NOMENCLATURE …………….xxii
CHAPTER Ⅰ INTRODUCTION ……….1
1.1 Motivation ………………1
1.2 Literature Survey ……3
1.2.1 Trajectory Interpretation of Quantum Mechanics and Multi-Path Phenomenon …………….3
1.2.2 Quantum Chaos Based on Quantum Trajectories …………………………..5
1.2.3 Probability from Classical Chaos ……..8
1.3 Contributions …………………………….8
1.4 Organizations ……………………………11
CHAPTER Ⅱ COMPLEX-TRAJECTORY INTERPRETATION OF
QUANTUM MECHANICS: THEORY AND PHENOMENON …...14
2.1 Quantum Hamilton-Jacobi Theory ……………15
2.2 Multi-Path Behavior from the Uncertainty of Imaginary Parts ………………….18
2.3 An Example: Complex Trajectories of Quantum Harmonic Oscillators in One-Dimensional oordinate Space ……………………………...20
2.4 Comparison with Bohmian Mechanics ……….22
2.5 Discussions ……………………………24
CHAPTER Ⅲ PARAMETERIZATION OF ALL PATH INTEGRAL
TRAJECTORIES …..………….25
3.1 The Fundamental Concepts of Path Integrals ……………………………………25
3.2 Entangled Free-Particle Trajectories in Complex Domain ………………………30
3.3 Parameterization of Path-Integral Trajectories …………………………………..35
3.4 Parameterization of Action ……………40
3.5 Evaluation of Path Integral …………46
3.6 Discussions ………………………………54
CHAPTER Ⅳ STRONG CHAOS IN ON-DIMENSIONAL
QUANTUM SYSTEM …… ……………………56
4.1 Quantum Trajectories within Complex Space …………………………………...56
4.2 Quantum Chaos in Harmonic Oscillator ……..59
4.3 Lyapunov Exponent …………………………..65
4.4 Strong Chaos ……………………………….69
4.5 Discussions …………………………………76
CHAPTER Ⅴ QUANTIFYING CHAOS: DETERMINE LYAPUNOV
EXPONENT FROM TIME SERIES DATA ………78
5.1 Time series of Dynamical Variables …….79
5.2 Phase Space Reconstruction ………..81
5.3 Estimation of Dimension …………...83
5.4 Approximate Jacobian Matrix : The Tangent Map ………………………………92
5.5 Lyapunov Exponents Determined from the Tangent Map ………………………95
5.6 The Choice of Time Delays ……….96
5.7 Construction Steps …………….97
CHAPTER Ⅵ PDF RECONSTRUCTION FROM CHAOTIC QUANTUM TRAJECTORIES ………………102
6.1 The Quantum System for a Charged Anisotropic Harmonic Oscillator in a Uniform Magnetic Field ………………………………………..104
6.2 Quantum Motion in the 2D Complex Space ….107
6.3 Quantum Chaos in a 2D Charged Anisotropic Harmonic Oscillator …………..108
6.3.1 Ground State( ) ……….108
6.3.2 The First Excited State ( ) ………...110
6.3.3 Multiple Paths in 2D Time-Independent Complex System ………………115
6.3.4 The Origin of Chaos: Quantum Potential ………………………………...118
6.4 Probability Interpretation from Chaos in Quantum System ……………………124
6.4.1 Probability in Quantum Mechanics ………124
6.4.2 The Probability Interpretation from Chaos ………………………………127
6.4.3 Ergodic Behavior ………………129
6.4.4 The Probability Distribution Associated with a Quantum Motion ……….130
6.4.5 The Probability Determined from the Statistics of Duration …………….134
6.4.6 Connection Between Lyapunov Exponent and Probability Distribution ...140
6.5 The Comparison with Bohmian Mechanics ……150
6.6 Discussions ……………..159
CHAPTER Ⅶ CONCLUSIONS AND FUTURE WORKS …….164
REFERENCES …………………166
PUBLICATION LIST ………………….174
VITA ………………………….175
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