透過波姆力學(Bohmian mechanics)這個具有因果內涵的量子理論，前人替流體力學形式的量子力學打下基礎。在波姆力學的架構底下，薛丁格方程式可以被拆解成兩個量子流體力學最基本的方程式。由於量子流體力學為吾人提供了熟悉的物理圖像，所以儘管這個架構有瑕疵仍然受到廣泛的討論。根據計算方式的不同，量子流體力學的研究可以分成解析型以及合成型兩種。解析型研究主要以提供詮釋為主，而合成型則和數值計算有關。在本論文中吾人將從另一更為完善且具因果內涵的量子力學─複數力學的角度出發，建立起新的量子流體力學，期盼能提供解析型以及合成型研究的一個新研究方向。在複數力學的架構之下，吾人建立了量子位流理論，將機率流和位勢流理論連結，提供解析型研究新的詮釋。而吾人同時也將舊有的數值計算技巧沿用到複數力學的架構之下，提供合成型研究一個新的視野。

The foundation of hydrodynamic formulation of quantum mechanics has been established by Bohm through his deterministic quantum theory, called Bohmian mechanics. Under the framework of Bohmian mechanics, Schrödinger equation has been separated into two equations, which are the basic formulation of quantum hydrodynamics. Because of the considerable insight that was provided by the hydrodynamics quantum mechanics, it has still been used for a diverse spectrum of physical processes even if it has several drawbacks. Depending on how they are computed, the methods may be broadly divided into two categories, analytic approach and synthetic approach. The purpose of analytic approach is not to solve the Schrödinger equation but rather to provide physical insights. The synthetic approach has something to do with the computational tool for solving the quantum hydrodynamic formulation.
The main purpose of the present thesis is to implement the new hydrodynamic formulation of quantum mechanics, which is constructed by a justified deterministic quantum theory, called complex mechanics. Under the scheme of complex mechanics, we established the quantum potential flow theory, which reveal a novel analogy between the probability flow and the potential flow, and we also incorporate the computational techniques into the new framework.

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