閉迴路導引系統形成波函數 之複數狀態空間動態，而量子算符便自然地從中而生

量子力學中的隨機最佳導引律
論文名稱(英文)	The Stochastic Optimal Guidance Law in Quantum Mechanics
校院名稱	成功大學
系所名稱(中)	航空太空工程學系
系所名稱(英)	Department of Aeronautics & Astronautics
學年度	102
學期	1
出版年	102
研究生(中文)	鄭烈烈
研究生(英文)	Lieh-Lieh Cheng
學號	P48951044
學位類別	博士
語文別	英文
論文頁數	102頁
口試委員	指導教授-楊憲東 口試委員-張書銓 口試委員-王大中 口試委員-李君謨 口試委員-孔健君 口試委員-黃榮興
中文關鍵字	隨機控制  最佳導引  量子軌跡  前導波
英文關鍵字	Stochastic Control  Optimal Guidance  Quantum Trajectory  Pilot Wave
學科別分類
中文摘要	延續德布洛伊導引波的概念，本論文將量子力學視為隨機最佳導引律的問題。在量子世界當中，吾人可將電子看做一架被導引的飛行器，而伴隨其身的導引波正是在追隨由維納程序所驅動的隨機標靶時並且同時在最小化剩餘成本函數考量之下，所設計出來的導引律。在藉由動態規劃法求解隨機最佳導引問題之後，吾人指出導引粒子運動的最佳導引波，其實就是薛丁格方程式的解：波函數 。同時間，吾人發現閉迴路導引系統形成波函數 之複數狀態空間動態，而量子算符便自然地從中而生。文章末吾人將求解在最佳導引律作用之下的量子軌跡，並顯示其在實數空間中的統計分佈與機率密度函數 的預測一致。
英文摘要	Following the de Broglie’s idea of pilot wave, this dissertation treats quantum mechanics as a problem of stochastic optimal guidance law design. The guidance scenario considered in the quantum world is that an electron is the flight vehicle to be guided and its accompanying pilot wave is the guidance law to be designed so as to guide the electron to a random target driven by Wiener process, while minimizing a cost-to-go function. After solving the stochastic optimal guidance problem by differential dynamic programming, we point out that the optimal pilot wave guiding the particle’s motion is just the wavefunction , a solution to Schrödinger equation; meanwhile, the closed-loop guidance system forms a complex state-space dynamics for , from which quantum operators emerge naturally. Quantum trajectories under the action of the optimal guidance law are solved and their statistical distribution is shown to coincide with the prediction of the probability density function .
論文目次	摘要……….… II ABSTRACT.... III CHINESE ABSTRACT OF EACH CHAPTER IV 誌謝……….… XIII CONTENTS……...…………………………………………… ………….XV LIST OF FIGURES XVII NOMENCLATURE XXI CHAPTER 1 INTRODUCTION 1 1.1 INCEPTION OF PILOT WAVE 1 1.2 BOHM’S PROPOSAL OF QUANTUM GUIDANCE 2 1.3 QUANTUM BROWNIAN MOTION 4 1.4 CONTRIBUTIONS 6 1.5 ORGANIZATION 7 CHAPTER 2 STOCHASTIC OPTIMAL CONTROL 9 2.1 VARIATIONAL APPROACH TO OPTIMAL CONTROL 9 2.2 DYNAMIC PROGRAMMING APPROACH TO OPTIMAL CONTROL 10 2.3 RANDOM WALK AND WIENER PROCESS 12 2.4 STOCHASTIC OPTIMAL CONTROL 16 2.5 OPTIMAL CONTROL OF A 1D STOCHASTIC SYSTEM 19 CHAPTER 3 OPTIMAL QUANTUM GUIDANCE LAW 21 3.1 TREATING QUANTUM MECHANICS AS A PURSUIT-EVASION GAME............21 3.2 OPTIMAL QUANTUM GUIDANCE LAW 25 3.3 RELATION TO QUANTUM BROWNIAN MOTION 29 CHAPTER 4 QUANTUM HAMILTON MECHANICS 31 4.1 QUANTUM HAMILTON EQUATIONS IN CLOSED GUIDANCE LOOP 31 4.2 RELATIONS TO QUANTUM OPERATORS 33 4.3 QUANTUM POTENTIAL FLOW 35 CHAPTER 5 QUANTUM TRAJECTORY UNDER OPTIMAL GUIDANCE LAW 42 5.1 SOLVING STOCHASTIC DIFFERENTIAL EQUATION 42 5.2 TRAJECTORY INTERPRETATION OF UNCERTAINTY PRINCIPLE 43 5.3 RECONSTRUCTING PDF BY OPTIMAL QUANTUM TRAJECTORIES IN THE COHERENT STATE 47 5.4 IMPROVING THE NUMERICAL ACCURACY OF OPTIMAL TRAJECTORY….. 53 5.5 RECONSTRUCTING PDF BY DIFFERENT INITIAL DISTRIBUTIONS…….. 58 5.5.1 UNIFORM DISTRIBUTION…….. 59 5.5.2 NORMAL DISTRIBUTION…….. 63 5.6 TESTING WITH FREE GAUSSIAN WAVE PACKET…….. 67 5.7 COMPARISON WITH BOHM'S GUIDANCE LAW 82 CHAPTER 6 CONCLUSIONS AND FUTURE WORK 94 6.1 CONCLUSIONS 94 6.2 FUTURE WORK 95 REFERENCES 97 PUBLICATION LIST 101 VITA…………………………………………………………………….....102
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