機率理論與軌跡理論是以不同觀點描述量子運動。機率論假設一個觀察者站在一個固定的空間點不動，觀察粒子通過此點的頻率。軌跡理論描述的是觀察者跟隨粒子移動，隨著時間不同，會有不同的粒子軌跡留下。隨著以軌跡觀點來描述量子系統的重要性日益增加，本論文希望能統整以統計學為基礎的軌跡觀點和複數力學框架下所建構的複數軌跡。其中，複數力學的量子軌跡，是由隨機最佳控制理論所推導出來的隨機運動方程式。而統計力學的量子軌跡隨機運動方程式，是由佛可-普朗克方程所求得。本文會以兩種力學所推導的隨機運動方程式，和量子系統機率密度進行比較。並根據波爾的相對性原理，當量子數趨近於無窮大時(n→∞時)，量子現象必須收斂到古典力學的巨觀行為。本論文發現複數力學的隨機運動方程式能描述相對性原理，而統計力學當n→∞的結果，並不能遵循牛頓運動軌跡的分布。這結果也證明了，粒子實際移動在複數空間之內，而不是我們通常認為的實數空間。

The probability theory and trajectory theory describe the quantum motion with different viewpoints. The probabilism assumes a viewer not moving along with a particle but standing on a fixed spatial point, who observes the frequency at which different particles pass him. The trajectory theory describes a viewer moving along with a particle, who passes different spatial points at different times. With the increasing importance of the trajectory-based interpretation of quantum mechanics, this thesis aims to integrate the trajectory view point with the statistical viewpoint under the framework of complex mechanics. The quantum trajectories in complex mechanics are derived by the theory of optimal stochastic control, which are complex random trajectories. On the other hand, the quantum trajectories in statistical mechanics are derived by the Fokker-Planck equation, which are real random trajectories. In this thesis, both random trajectories solved from complex mechanics and statistical mechanics are discussed, and the convergence of their spatial distributions to the probability density function is compared. According to the Bohr’s corresponding principle, the quantum behavior, as quantum number approaches to infinity (n→∞), must converge to the macroscopic behavior described by classical mechanics. The major finding of this thesis is that only the complex random trajectories satisfy the Bohr’s corresponding principle, while the real trajectories solved from statistical mechanics fail to follow the Newtonian trajectories as n→∞. This result provides obvious evidence that particles actually move in a complex space, but not in a real space as we think commonly.

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