幾何相位是在量子力學發展後期才開始漸漸被查覺，因為複數的波函數在絕熱狀態下做封閉路徑的運動必須滿足單一性，所以幾何相位在傳統量子力學中產生了。但由於傳統機率詮釋的量子力學中缺乏動態軌跡的概念，所以在於驗證這幾何相位的適用性上有較大的缺陷。複數力學，做為巨觀古典力學以及微觀量子力學之間的橋梁，賦予了量子力學所不足的非線性動態軌跡概念。
由於複數非線性動態的加入，使得經典力學中分析非線性系統的工具得以導入使用，讓量子力學可以不再完全以機率來詮釋，它可以在複數空間中呈現另一種動態軌跡的面貌。複數力學對於完美詮釋量子系統的非線性複數薛丁格方程式重新解讀，發現量子系統其實是必須建構在複數空間之中的，並且在這複數空間中的量子系統必須導入一個量子位勢，使其滿足複數薛丁格方程式。
本論文以複數力學所提供的非線性動態軌跡出發，導入古典力學中非線性系統的指標定理，重新定義出屬於量子力學在複數空間中的量子指標定理，並且找出相對應的指標以及相對應的物理意義，而幾何相位就是量子指標定理中的一種重要的幾何意義。對於波函數的幾何相位，在複數力學架構下可以用複數空間中的動態封閉軌跡直接計算證明，而其理所當然的滿足傳統量子力學中所強調的波函數單一性，完整解釋了傳統量子力學中幾何相位的部分。藉由量子指標定理在幾何相位上的計算與模擬，將傳統量子力學中的幾何相位與複數力學中的量子幾何相位緊密的結合。論文中的結果顯示兩者之間的相依特性，並且以電子在均勻磁場中的動態與幾何相位相關的Aharonov-Bohm效應實驗做比較。不僅如此，在論文中與許多實驗結果做比對，一再的證明量子幾何相位與幾何相位之間的等價關係。
最後，作者將利用巨觀尺度下擁有量子行為的超導體元件做為量子幾何相位在工程領域的應用。約瑟夫森接面在超導體工程元件中是最基本的，它使用了巨觀尺度下同調的相位作為變數來描述特性，這與量子幾何相位的描述不謀而合。在複數力學的架構中，超導量子工程元件上的應用，相較於傳統量子力學的機率描述，多了份較直覺的粒子軌跡詮釋，在分析應用上更加的方便與簡潔。

The geometrical phase is noticed after the development of conventional quantum mechanics gradually matured. It describes the single-value property of the wavefunction phase accumulation when varying adiabatically around a closed circuit. However, due to the lack of the concept of a dynamic trajectory in the probability-interpreted quantum mechanics, it is difficult to verify the applicability of the geometric phase. Complex mechanics, as a bridge between the macroscopic classical mechanics and the microscopic quantum mechanics, provides a nonlinear dynamic trajectory interpretation that cannot be revealed in conventional quantum mechanics.
The nonlinear system analysis tools used in classical mechanics now can be applied to the quantum world via complex nonlinear dynamic representation. Hence, the quantum world can be displayed in terms of a dynamic trajectory interpretation instead of using probability interpretation. Complex mechanics provides the complex Schrödinger equation with a new description by considering the complex space extension and a quantum potential within this complex space.
The dissertation begins with the introduction of the nonlinear dynamic trajectory on the basis of complex mechanics, than applies the index theorem in classical nonlinear systems to the quantum system. This is done by presenting the quantum index theorem and also is able to find the corresponding physical meanings for new defined quantum indices. It is found that the geometrical phase of the wavefunction gives an important geometrical meaning in the quantum index theorem. It shows that the closed loop dynamic trajectory in complex space satisfies the single-value property of the wavefunction and classically exhibits the geometrical phase which originally could not be explained properly in conventional quantum mechanics. The quantum geometrical phase interpreted by the dynamic trajectory has the coherent appearance of the geometrical phase proposed by probability-interpreted quantum mechanics. This coherence relation is discussed via the study of the electronic dynamics in a uniform magnetic field and its related Aharonov-Bohm effect. The simulation result of the study is compatible with the experimental result, which provides strong evidence that the trajectory interpretation in complex space is indeed a correct framework of quantum mechanics.
Finally, the author actually applies the quantum geometrical phase to the field of engineering via a superconducting device which has macroscopic quantum behavior. Josephson junction is a basic device used in superconducting engineering. The characteristics of the junction are described by a fictitious particle’s tunneling behavior on the wavefunction phase difference plane in the macroscopic scale. Complex mechanics connects quantum mechanics to classical mechanics, providing the dynamic trajectory interpretation such that it can discuss and clarify those problems which are not represented fully in quantum mechanics.

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