本論文的目的即是在為量子系統建立一個狀態空間描述法，透過最佳化隨機控制來驗證量子系統其實就是最佳化的隨機系統。由於最佳化的過程包含兩個步驟:複數化和隨機化，這結果顯示量子運動實際上是發生在複數空間的一種隨機運動。本論文推導出量子隨機運動所需滿足的隨機微分方程式，並且加以套用在氫原子的量子動態、角動量及雙原子分子中的量子動態；再透過方程式的求解，獲得貼近真實世界的量子隨機軌跡。

The purpose of this paper is to establish a state-space for quantum systems, according to optimal stochastic control, and to verify that a quantum system is actually an optimized stochastic system. The proposed optimization process comprises two steps: complexification and randomization, and the outcome of the process shows that a quantum motion is actually a complex Brownian motion. After our study, it becomes clear that a quantum path is random and fractal, and is governed by a stochastic differential equation, from which a quantum paths can be solved for electronic motions in hydrogen atom and for orbital angular momentum and quantum dynamics in diatomic molecules.

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