奈米結構量子點歷來素有人造原子之稱，然而不像一般的天然原子，量子點中仍然有一些現象是近代量子物理學所無法解釋的，在低磁場強度下的磁電阻(magneto-resistance)現象就是其一。從古典力學的角度來看，磁電阻現象可由電子在量子點中的共振散射軌跡來解釋，然而牛頓力學的世界中並不存在量子態與波函數，這使得古典力學的軌跡詮釋雖然可以獲得些許成果，卻與真實的奈米世界格格不入。另一方面從量子力學的角度來看，因為電子的行為只有機率分布沒有軌跡，以致無法建立軌跡共振與磁阻現象之關聯性。本文從複數力學的觀點來解釋量子點中的磁電阻現象，複數力學作為古典與量子世界的橋樑，可以從古典力學所缺乏的量子化觀點，來提供磁電阻現象一個量子軌跡上的解釋。相對於國內外相關研究，本論文所提供的量子點非線性動力學的分析方法是先前文獻所沒有的，它不僅可應用於磁電阻現象的分析，也可應用於量子點的其他光、電、熱等諸效應上。複數力學在這裡的主要功能是提供量子運動的動態方程式，這是電子軌跡分析與量子渾沌分析所需要知道的資訊，但卻是經典量子力學所無法提供的。有了複數力學這個介面，所有應用古典力學的分析方法來解釋磁阻現象的概念都可以完整地結合量子化現象而應用在量子點的分析中。
本文首先建立電子於等向性量子點(圓形)內的非線性動態方程式。藉由求解此運動方程式，吾人證實複數軌跡的封閉性正是造成磁電阻現象的主要原因。電子進入量子點後在固定且封閉的路徑上運行且產生幾何共振現象，準確地回到原來的出發點。當外加磁場強度值恰好滿足軌跡的共振條件時，電子的運動將完全被限制在量子點的入口附近，使其無法離開量子點而造成了電阻的上升。文中的研究成果顯示，計算所得之共振磁場強度與造成電阻峰值之磁場強度測量值一致。
接下來吾人透過複數力學進一步建立非等向性量子點的模型(橢圓形)，使其更接近實際量子點的行為。對於橢圓形量子點中出現的渾沌現象，吾人利用李氏指數來量化並計算電子軌跡的混沌強度，並且分析電子混沌動態對於磁電阻現象的影響。
最後吾人在本論文中求出電子在不同量子態的混沌軌跡，利用一條量子混沌軌跡在空間所掃過軌跡點的分布與機率密度函數所預測者一致的特性，重建量子力學中的機率密度函數。並且證實渾沌程度越強的電子軌跡，重建出來的機率密度函數越完整。

Nanostructure quantum dot is regarded as the well-known “artificial atom” in recent years. However, unlike common atoms, quantum dots still have some characteristics which still remain unexplainable by conventional quantum mechanics. Magneto-resistance under the action of low-intensity magnetic field is one of these unaccountable phenomena. From the viewpoint of classical mechanics, the effect of trajectory resonance might account for the occurrence of magneto-resistance in quantum dots. However, because classical trajectories cannot exhibit the quantum-state-dependent nature of quantum dots, only qualitative description of magneto-resistance could be made by using classical approaches. On the other hand, the absence of trajectory interpretation in quantum mechanics makes it impossible to relate magneto-resistance to the resonance of quantum trajectories. To conquer the existing difficulty, the present thesis provides a nonlinear dynamic analysis of magneto-resistance via the framework of complex mechanics, which endows classical mechanics with quantization and furnishes quantum mechanics with dynamics equations of motion. For a single quantum dot, the quantum dynamical equations of motion established in this thesis, which have not been reported in the existing literature, are indispensable to the study of the nonlinear phenomena, such as stability, bifurcation and chaos in quantum dots. Through the interface of complex mechanics, magneto-resistance as well as other optic, electronic and thermal properties of quantum dots can be analyzed quantum mechanically by using the tools and concepts already developed in classical mechanics.
At first a set of nonlinear equations of motion are established to describe the electronic quantum motion in an isotropic (circular) quantum dot. In terms of the solutions to the equations of motion, we reveal that the closeness of the complex trajectories is the main cause of magneto-resistance in quantum dots. An incident electron with a closed trajectory implies that it returns to the entrance of the quantum dot and thus raises the electronic resistance of the dot. We point out that the existence of such closed trajectories is a consequence of geometric resonance. The critical magnetic field leading to geometric resonance is derived and a preliminary result shows that the computed critical magnetic field is identical to the observed magnetic field causing magneto-resistance peaks.
We then consider an anisotropic quantum dot with elliptic shape, which is closer to the practical situation than a circular dot. Quantum chaotic trajectories in various quantum states are found out first and the distribution of the spatial points swept by a single quantum trajectory is shown to be consistent with the probability distribution determined by wavefunctions. We make use of Lyapunov exponent to quantify the degree of chaos of quantum trajectories. Our study reveals a remarkable observation that chaotic behavior is favorable for an electron to pass the dot and hence contributes to the conductance of the quantum dot.

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