古典系統同時具備狀態空間描述(state-space description)與波函數(或稱作用量函數)描述，其中狀態空間描述又稱內部描述(internal description)，是由Hamilton運動方程式呈現系統內部的運作機制；波函數描述又稱外部描述(external description)，用以說明輸入與輸出間之關係，是由Hamilton-Jacobi方程式的解所提供。然而對於量子系統而言，目前只有波函數描述法，並沒有相對應的量子狀態空間描述法。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。古典系統的狀態空間描述與波函數描述，可以由最小作用量原理同時推導出來。本論文指出，量子系統的狀態空間描述與波函數描述，依然可以由最小作用量原理同時推導而得，只要吾人將粒子的運動從實數擴展到複數領域，並加入一個隨機擾動量。經由複數化與隨機化的擴增，最小作用量原理的求解變成一個最佳化隨機控制的問題。在最佳控制法則的作用下，吾人證明受控系統的閉迴路模式便是所要求的量子系統狀態空間表示式。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。本論文推導出複數隨機運動所需滿足的隨機微分方程式，再透過此方程式的求解，獲得貼近真實世界的量子隨機軌跡。在此之前，量子隨機軌跡只能在實驗室中獲得。在指定的解析度下，本論文求出量子隨機軌跡，並證實其軌跡點的空間分布與波函數的預測一致；吾人同時計算出量子隨機軌跡的碎形維度為2，此結果也與理論的預測一致。

Classical systems can be described either by an internal (state-space) model, which is a set of Hamilton equations of motion, or by an external (input-output) model, which is a wavefunction solved from the Hamilton-Jacobi equation. However, so far a quantum system can only have the external representation as a wavefunction . The aim of this dissertation is to establish a state-space representation for . We recall that the state-space model and the wavefunction model for a classical system can be derived simultaneously by the principle of least action. We will see that this result is still true for a quantum system, if the coordinate is extended to the complex domain and superposed by a noise. The principle of least action with such an extension yields an optimal stochastic control problem. Once an optimal control law is found, the resulting closed-loop model provides a state-space realization for the quantum system. The 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 path can be solved and visualized with any given resolution.

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