臺灣博碩士論文加值系統

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。

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.

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

[1]Antoniou, I., and Gustafson, K., 1997, “From Irreversible Markov Semigroups to Chaotic Dynamics”, Physica A, Vol. 236, pp.296-308.

[2]Antoniou, I., and Gustafson, K., 1993, “From Probabilistic Description to Deterministic Dynamics”, Physica A, Vol. 197, pp.153-166.

[3]Antoniou, I., Gustafson, K. and Suchanecki, Z., 1998, “On the Inverse Problem of Statistical Physics: from Irreversible Semigroups to Chaotic Dynamics”, Physica A, Vol. 252, pp.345-361.

[4]Antoniou, I., Christidis, TH. and Gustafson, K., 2004, “Probability from Chaos”, International Journal of Quantum Chemistry, Vol. 98, pp.150-159.

[5]Arfken, G., 1985, “Hypergeometric Functions”, §13.5 in Mathematical Methods for Physicists, 3rd ed. Academic Press, pp.748-752.

[6]Balian, R., and Bloch, C., 1970, “Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: I. Three-dimensional Problem with Smooth Boundary Surface”, Annals of Physics, Vol. 60, No. 2, pp.401-447.

[7]Balian, R., and Bloch, C., 1971, “Asymptotic Evaluation of the Green's Function for Large Quantum Numbers”, Annals of Physics, Vol. 63, No. 2, pp.592-606.

[8]Balian, R., and Bloch, C., 1971, “Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain. II. Electromagnetic Field. Riemannian Spaces”, Annals of Physics, Vol. 64, No. 1, pp.271-307.

[9]Balian, R., and Bloch, C., 1972, “Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: III. Eigenfrequency Density Oscillations”, Annals of Physics, Vol. 69, No. 1, pp.76-160.

[10]Balian, R., and Bloch, C., 1974, “Solution of the Schrödinger Equation in Terms of Classical Paths”, Annals of Physics, Vol. 85, No. 2, pp.514-545.

[11]Barut, A.O., and Duru, I.H., 1988, “Path Integration via Hamilton-Jacobi Coordinates and Applications to Potential Barriers”, Physical Review A , Vol. 38, pp.5906-5909.
[12]Benettin, G., and Strelcyn, J.M., 1978, “Numerical Experiments on the Free Motion of a Point Mass Moving in a Plane Convex Region: Stochastic Transition and Entropy”, Physics Review A, Vol. 17, pp.773-785.

[13]Berry, M., 1989, “Quantum Chaology, not Quantum Chaos”, Physica Scripta, Vol. 40, pp.335-336.

[14]Blümel, R., and Smilansky, U., 1990, “Quantum Mechanical Suppression of Chaos”, Physics World 3 (2), pp.30-34.

[15]Bogojevic, A., Balaz, A., and Belic, A., 2005, “Jaggedness of Path Integral Trajectories”, Physics Letters A, Vol. 345, pp.258-264.

[16]Bohigas, O., Giannoni, M.J., and Schmit, C., 1984, “Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws”, Physics Review Letter, Vol. 52, pp.1-4.

[17]Bohm, D., and Vigier, J.P., 1954, “Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations”, Physics Review, Vol. 96, pp.208-216.

[18]Bohm D., and Hiley B.J., 1993, “The Undivided Universe”, Routledge, London.

[19]Bohm, D., 1952, “A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I”, Physics Review, Vol. 85, pp.166-179.

[20]Bohm, D., 1952, “A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II”, Physics Review, Vol. 85, pp.180-193.

[21]Burghardt, B., and Stolze, J., 1999, “Numerical Evaluation of Coherent-State Path Integrals in Quantum Dynamics”, Journal of Physics A: Mathematical and General, Vol. 32, pp.2075-2090.

[22]Casdagli, M., Sauer, T., and Yorke, J.A., 1991, “Embedology”, Journal of Statistical Physics, Vol. 65, pp.579-616.

[23]Chattaraj, P.K., and Sengupta, S., 1993, “Quantum Fluid Dynamics of a Classically Chaotic Oscillator”, Physics Letter A, Vol. 181, pp.225-231.

[24]Creutz, M., and Freedman, B., 1981, “A Statistical Approach to Quantum Mechanics”, Annals of Physics, Vol. 132, pp.427-462.

[25]de Alcantara Bonfim, O.F., Florencio, J., and Sá Barreto, F.C., 1998, “Chaotic Dynamics in Billiards Using Bohm’s Quantum Mechanics”, Physics Review E, Vol. 58, pp.2693-2696.

[26]de Polavieja, G.G., Borondo, F., and Benito, R.M., 1994, “Scars in Groups of Eigenstates in a Classically Chaotic System”, Physics Review Letter, Vol. 73, pp.1613-1616.

[27]de Sales, J.A., and Florencio, J., 2003, “Quantum Chaotic Trajectories in Integrable Right Triangular Billiards”, Physics Review E, Vol. 67, pp.016216.1 – 6.

[28]Dippel, O., Schmelcher, P., and Cederbaum, L.S., 1994, “Charged Anisotropic Harmonic Oscillator and The Hydrogen Atom in Crossed Fields”, Physics Review A, Vol. 49, pp.4415-4429.

[29]Eckmann, J.P., Oliffson Kamphorst, S., 1986, “Lyapunov Exponents from Time Series”, Physics Review A, Vol. 34, pp.4971-4979.

[30]Eckmann, J.P., and Ruelle, D., 1985, “Ergodic Theory of Chaos and Strange Attractors”, Reviews of Modern Physics, Vol. 57, pp.617-656.

[31]Elnaschie M.S., 1995, “On Conjugate Complex Time and Information in Relativistic Quantum Theory”, Chaos, Solitons and Fractals, Vol. 5(8), pp. 1551-1555.

[32]Elnaschie M.S., 2000, “On the Unification of the Fundamental Forces and Complex Time in the Space”, Chaos, Solitons and Fractals, Vol. 11, pp.1149-1162.

[33]Elnaschie, M.S., 2005, “The Feynman Path Integral and E-infinity from the Two-Slit Gedaneken Experiment”, International Journal of Nonlinear Science and Numerical Simulation, Vol. 6(4), pp.335-342.

[34]Elnaschie, M.S., 2005, “A New Solution for the Two-Slit Experiment”, Chaos, Solitons and Fractals, Vol. 25, pp.935-939.

[35]Elnaschie M.S., 2005, “Non-Euclidean Spacetime Structure and the Two-Slit Experiment”, Chaos, Solitons and Fractals, Vol. 26, pp.1-6.

[36]Elnaschie, M.S., 2006, “Hilbert Space, the Number of Higgs Particles and the Quantum Two-Slit Experiment”, Chaos, Solitons and Fractals, Vol. 27, pp.9-13.

[37]Feynman, R.P., and Hibbs, A.R., 1965, “Quantum Mechanics and Path Integrals”, MacGraw-Hill.

[38]Giannoni, M.J., Voros, A., and J. Zinn-Justin, 1991, “Chaos and Quantum PhysicsLes Houches Lecture Series 52”, North-Holland, Amsterdam.

[39]Goldstein, H., 1980, “Classical Mechanics”, Chapter 10, 2nd Ed., Addison- Wesley Publishing Company.

[40]Goldstein, S., Misra, B., and Courbage, M., 1981, “On Intrinsic Randomness of Dynamical Systems”, Journal of Statistical Physics, Vol. 25, pp.111-126.

[41]Goodrich, R., Gustafson, K., and Misra, B., 1980, “On Converse to Koopman's Lemma”, Physica A, Vol. 102, pp.379-388.

[42]Grassberger, P., and Procaccia, 1983, “Characterization of Strange Attractors”, Physics Review Letter, Vol. 50, pp.346-49.

[43]Gutzwiller, M.C., 1990, “Chaos in Classical and Quantum Systems”, Springer, Berlin.

[44]Haake, F., 1990, “Quantum Signatures of Chaos”, Springer, Berlin.

[45]Herold, H., Ruder, H., and Wunner, G., 1981, “The Two-Body Problem in the Presence of a Homogeneous Magnetic Field”, Journal of Physics B, Vol. 14, pp.751-764.

[46]Hilborn, R.C., 2000, “Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers”, Oxford university.

[47]Hilbert, D., 1902, “Mathematical Problems, Bulletin of the American Mathematical Society”, Vol. 8, pp.437-479.

[48]Hirsch, M.W., and Smale, S., 1965, “Differential Equations, Dynamic Systems and Linear Algebra”, Academic Press, New York.

[49]Holland, P. R., 1993, “The Quantum Theory of Motion”, Cambridge University.

[50]Hopf, E., 1934, “On Causality, Statistics, and Probability”, Journal of Mathematics and Physics, Vol. 13, pp.51-102.

[51]Izrailev, F.M., 1990, “Simple Models of Quantum Chaos: Spectrum and Eigenfunctions”, Physics Reports, Vol. 196, pp.299-392.

[52]Jahnson, R.A., Palmer, K.J., and Sell, G., 1987, “Ergodic Properties of Linear Dynamical Systems”, SIAM Journal on Mathematical Analysis, Vol. 18, pp. 1-33.

[53]Jancel, R., 1969, “Foundations of Classical and Quantum Statistical Mechanics”, Oxford, UK.

[54]José, J.V., 1988, “Quantum Manifestations of Classical Chaos: Statistics of Spectra,” in Hao, B.L., editor, Directions in chaos, Vol. 2, Singapore.

[55]Kleinert, H., 2004, “Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets”, World Scientific.

[56]Kolmogorov, A.N., 1960, “A New Metric Invariant for Transitive Dynamical Systems and Automorphisms in Lebesgue Spaces, Doklady of Russian Academy of Sciences”, Mathematical Reviews, Vol. 21, pp.386.

[57]Konkel, S., and Makowski, A.J., 1998, “Regular and Chaotic Causal Trajectories for the Bohm Potential in a Restricted Space”, Physics Letter A, Vol. 238, pp.95-100.

[58]Koopman, B., 1931, “Hamiltonian Systems and Transformations in Hibert Space”, Proceedings of the National Academy of Sciences of the United States of America, Vol. 17, pp.315-318.

[59]Koopman, B., and von Neumann, J., 1932, “Dynamical Systems of Continuous Spectra”, Proceedings of the National Academy of Sciences of the United States of America, Vol. 18, pp.255-266.

[60] Kroger, H., 2000, “Fractal Geometry in Quantum Mechanics, Field Theory and Spin Systems”, Physics Reports, Vol. 323, pp.81-181.

[61]Krylov, N.S., 1979, “Works on the Foundations of Statistical Physics”, Princeton University.

[62]Mané, R., Rand, D., and Young, L.S., 1981, “Dynamical Systems and Turbulence”, Springer, Berlin.
[63]Mehta, M.L., 1991, “Random Matrix Theory”, Academic Press.

[64]Meyn, S.P., and Tweedie, R.L., 2005, “Markov Chains and Stochastic Stability”, Springer-Verlag, London.

[65]Miller, S.C., and Good, R.H., 1953, “A WKB-Type Approximation to the Schrödinger Equation”, Physical Review, Vol. 91, pp.174-179.

[66]Misra, B., Prigogine, I., and Courbage, M., 1979, “From Deterministic Dynamics to Probabilistic Descriptions”, Physica A, Vol. 98, pp.1-26.

[67]Misra, B., and Prigogine, I., 1983, “In Long Time Predictions in Dynamical Systems”, Wiley, New York, pp.21-43.

[68]Parmenter, R.H. and Valentine, R.W., 1995, “Deterministic Chaos and the Causal Interpretation of Quantum Mechanics”, Physics Letter A, Vol. 201, pp.1-8.

[69]Parmenter, R.H., and Valentine, R.W., 1997, “Chaotic Causal Trajectories Associated with a Single Stationary State of a System of Noninteracting Particles”, Physics Letter A, Vol. 227, pp.5-14.

[70]Penrose, R., 1979, “Singularities and Time-Asymmetry, in S.W. Hawking and W. Israel, eds., General Relativity: An Einstein Centenary Survey”, Cambridge University.

[71]Pesin, Ya. B., 1977, “Lyapunov Characteristic Exponents and Smooth Ergodic Theory”, Russian Mathematical Surveys, Vol. 32, pp.55-114.

[72]Prigogine, I., 1980, “From Being to Becoming: The New Science of Connectedness”, San Francisco.

[73]Rabinovich, M.I., 1979, “Stochastic Self-Oscillations and Turbulence”, Soviet Physics Uspekhi, Vol. 21, pp.443-469.

[74]Ribeiro, A.D., de Aguiar, M.A.M., and Baranger, M., 2004, “Semiclassical Approximation Based on Complex Trajectories”, Physical Review E, Vol. 69, pp.066204.1-14.

[75]Rosenstein, M.T., Collins, J.J., and De Luca, C.J., 1993, “A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets”, Physica D, Vol. 65, pp.117-134.

[76]Schmelcher, P., and Cederbaum, L.S., 1993, “Two-body Effects of The Hydrogen Atom in Crossed Electric and Magnetic Fields”, Chemical Physics Letters, Vol. 208, pp.548-554.

[77]Seligman, T.H., Verbaraarschot, J.J.M., and Zirnbauer, M.R., 1984, “Quantum Spectra and Transition from Regular to Chaotic Classical Motion” Physics Review Letter, Vol. 53, pp.215-217.

[78]Shankar, R., 1994, “Principles of Quantum Mechanics”, Chapter 7, 2nd Ed., Plenum Press.

[79]Sharlow, M.F., 1998, “A New Non-Hausdorff Spacetime Model for Resolution of the Time Travel Paradoxes”, Annals of Physics, Vol. 263, pp.179-197.

[80]Sinai, Y.G., 1988, “About A. N. Komogorov’s Work on the Entropy of Dynamical Systems, Ergodic Theory and Dynamical Systems”, Vol. 8, pp.501-502.

[81]Takens, F., 1980, “Detecting Strange Attractor in Turbulence”. In D. Rand and L. S. Young, editors, Dynamical Systems and Turbulence, Warwick, page 366, Springer, Berlin, 1981.

[82]Visser, M., 1995, “Lorentzian wormholes: From Einstein to Hawking”, AIP Press, Woodbury, N.Y..

[83]Von Plato, J., 1991, “Boltzmann's Ergodic Hypothesis”, Archive for History of Exact Sciences, Vol. 42, pp.71-89.

[84]Yang, C.D., 2005, “Wave-Particle Duality in Complex Space”, Annuals of Physics, Vol. 319, pp.444-470.

[85]Yang, C.D., 2005, “Quantum Dynamics of Hydrogen Atom in Complex Space”, Annuals of Physics, Vol. 319, pp.399-443.

[86]Yang, C.D., 2006, “Modeling Quantum Harmonic Oscillator in Complex Domain”, Chaos, Solitons and Fractals, Vol. 30, pp.342-362.

[87]Yang, C.D., 2006, “Quantum Hamilton Mechanics: Hamilton Equations of Quantum Motion, Origin of Quantum Operators, and Proof of Quantization Axiom”, Annals of Physics, Vol. 321, pp.2876-2926.

[88]Yang, C.D., 2007, “Quantum Motion in Complex Space”, Chaos, Solitons and Fractals, Vol. 33, pp.1073-1092.

[89]Yang, C.D., 2008, “Trajectory Interpretation of the Uncertainty Principle in 1D Systems Using Complex Bohmian Mechanics”, Physics Letters A, Vol. 372, pp.6240-6253.

[90]Zakrzewski, J., Dupret, K., and Delande, D., 1995, “Statistical Properties of Energy Levels of Chaotic Systems: Wigner or Non-Wigner?”, Physics Review Letter, Vol. 74, pp.522-525.

[91]Zurek , W.H., and Paz, J.P., 1994, “Decoherence, Chaos, and the Second Law”, Physics Review Letter, Vol. 72, pp. 2508-2511.