我們利用一個雙粒子系統模型來研究量子布朗運動中的動力學。由於我們的系統是一種開放量子系統，所以我們選用Feynman與Vernon在路徑積分版本的量子力學中發展出來的作用泛函方法來處理它；這個方法被稱為Feynman-Vernon理論。在推導出傳播子與主方程式之後，我們可以用約化密度矩陣來表示系統狀態，並且能夠知道系統狀態是如何演化到任何時間點。透過約化密度矩陣，我們能計算出各種物理量的期望值，特別是系統位置變異數與動量變異數的期望值，這兩個期望值正是探討不確定性原理的起點。本研究工作主要集中在雙粒子系統的約化密度矩陣與主方程式的詳細推導。未來將可以使用約化密度矩陣來計算任何感興趣的物理量。

We take a two-particles system as our model to study the dynamics of quantum Brownian motion. Because of our system is an open quantum system, we use the Feynman-Vernon theory, which is the influence-functional method in the path-integral formulism, to deal with it. After deriving the propagator and the master equation, we evolve the system to any time, and expressing system state by a reduced density matrix. From that, we calculate the expectation values for the variances of position x and momentum p, which is the starting point to study the generalized uncertainty relation at finite temperature. This work mainly focuses on the detailed derivation of the reduced density matrix for two-particle system and the master equation of that system. One can then use the reduced density matrix to calculate any observable as we want.

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