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研究生：	洪子超 Hung, Tzu-Chao
論文名稱：	以費雪訊息探討量子理論 Fisher information in Quantum Theory
指導教授：	楊緒濃 Nyeo, Su-Long
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2014
畢業學年度：	102
語文別：	英文
論文頁數：	49
中文關鍵詞：	費雪訊息、統計距離、本徵的機率分佈、薛丁格方程式
外文關鍵詞：	Fisher information, statistical distance, intrinsic probability distributions, Schrodinger equation
相關次數：	點閱：150 下載：1
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費雪訊息 (Fisher Information) 在統計量測和訊息理論中扮演著重要角色。當我們量測系統時引發出類似機率概念的東西，稱之為「似然性」，而其可描述一系統之亂度。費雪訊息與 Kullback-Leibler 熵值相關，且滿足 Cramer-Rao 不等式，更與統計距離 (Statistical Distance) 有密切關連。

在此篇論文中，藉由費雪訊息我們提供連接古典物理和量子物理的橋梁。而藉著最小費雪訊息原理 (The Principle of Minimum Fisher Information)，我們可以從哈密頓-亞可比方程式 (Hamilton-Jacobi Equation) 得到量子物理 --- 薛丁格方程式。

費雪訊息矩陣 (Fisher Information Matrix) 可被視為一度規張量(Metric) g，我們由此可導出於時空中機率分部的限制條件。我們假設時空存在著本徵的機率分佈，藉由著優化費雪訊息度規 (Fisher Information Metric) 可以得到機率分佈。藉由此方法我們可得到一些微分方程來描述在時空中本徵的機率分佈。

Fisher information, which is an important concept in statistical estimation theory and information theory, provides a measurement of a disorder system, which is specified by a corresponding probability, the likelihood. Fisher information is related to the Kullback-Leibler entropy, satisfies the Cramér-Rao inequality, and is also related to statistical distance.

In this thesis, we provide a bridge to connect classical and quantum mechanics by using Fisher information. Following the principle of minimum Fisher information, we describe the mechanism of quantum world — the Schrödinger equation from the Hamilton-Jacobi equation.

For a given metric gµν, which is identified as Fisher information metric, we generate new constraints for the probability distributions for physical systems.We postulate the existence of intrinsic probability distributions for physical systems, and calculate the probability distribution by optimizing the Fisher information metric under specified constraints. Accordingly, we get differential equations for the probability distributions.

abstract i
摘要 ii
謝誌 iii
Statistical Methods and Estimation   1
1 Measurement   2
2 Fisher Information   3
3 Cramér-Rao Bound   4
4 Relation of Fisher Information to Kullback-Leibler Entropy and Fisher Information Matrix   7
4.1 Relation of Fisher Information to Kullback-Leibler Entropy   7
4.2 Fisher Information Matrix   8

Statistical Distance   10
1 Definition of Statistical Distance   10
2 Relation to the Fubini-Study Metric   13
3 Classical Distinguishability Metric   14

The Bridge between Classical and Quantum Mechanics   18
1 Introduction   18
2 Probability Distributions for Physical Systems   19
3 The Dimension of Fisher Information   21
4 Time-Independent Schrödinger Equation   24
5 Relation between Probability Distribution and Action  26
6 Time-Dependent Schrödinger Equation   28
7 Klein-Gordon Equation   28
8 Conclusion   30

Probability Distribution in Schwarzschild Geometry   31
1 The Fisher Information Metric   31
2 Probability Distributions in Minkowski Space   32
2.1 Minkowski space in Cartesian coordinates   32
2.2 Minkowski space in spherical coordinates   35
2.3 Connection to quantum physics   37
3 Probability Distribution in Schwarzschild Geometry   38
3.1 Is there any connection to quantum physics?   40
4 Conclusion   41
 
Appendix A Derivation of the TISE according to Brownian Motion   43
Appendix B Hydrogen Wave Function   45
Bibliography   47
                                    

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校內：2019-07-21公開
校外：2019-07-21公開

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