在複數力學(Complex Mechanics)的架構下，吾人從建立在複數空間的量子漢彌爾頓力學(Quantum Hamilton mechanics)出發，藉著所獲得的非線性動態方程式，不僅可描繪粒子在複數空間的量子軌跡，證實與正統量子力學機率預測相一致的雙原子分子量子軌跡是存在的，此外還能將古典非線性動態分析的方法引進量子世界中，這嶄新的方法提供了分析量子分岔(Bifurcation) 現象的數學工具。藉著分岔參數的逐步變化可直接觀察到平衡點與軌跡分佈所產生的劇烈變化。量子漢彌爾頓力學適用於各種座標系，除了卡式座標系外，也適用於探討雙原子分子時所採用的球座標系，用以模擬耦合振盪、旋轉與自旋的量子行為，分析引進角動量運動所產生的影響，並應用在分子的振動旋轉光譜與核自旋的分析上；當探討在扁平球座標系下的氫分子離子時，吾人證實在不同初始位置條件下所獲得的大量軌跡集合，等義於由原子軌域線性理論(LCAO)所構建成的分子軌域模型。

Under the framework of complex mechanics, we utilize the Hamilton mechanics to describe quantum trajectories represented in complex space by the nonlinear dynamic equations, and point out that quantum trajectories of a diatomic molecule completely are consistent with the probabilistic prediction of standard quantum mechanics. In addition, the nonlinear Hamilton equations establish a bridge from classic nonlinear dynamic system theory to quantum bifurcation, which allows the latter to be analyzed completely by the tools developed by the former. This new approach provides the necessary mathematic framework for the analysis of quantum bifurcation and makes it possible to identify quantum bifurcation by the direct evidence of the sudden change of fixed points and their surrounding trajectories. The quantum Hamilton mechanics has a coordinate-independent expression, which could not only work in cartesian coordinate, but also work in spherical coordinates for diatomic molecules and in prolate spheroidal coordinates for hydrogen molecular ion. In terms of quantum Hamilton mechanics, we could model the rotation-dependent vibrational dynamics and nuclear spin dynamics of diatomic molecules in spherical coordinates and analyze the influence of the angular quantum motion on the vibrational quantum motion. In addition, we have computed the vibration-rotation absorption transition from complex mechanics directly. Furthermore, the superposition of the resulting complex quantum trajectories generated from different initial conditions can reconstruct the molecular orbitals determined from the linear combination of atomic orbitals (LCAO) in hydrogen molecule ion.

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