本論文的重點在於利用量子流體力學的觀點研究奈米尺度下，雙孔(狹縫)噴流之組成、動態及量子能之變化。特別強調本論文定位是以哥本哈根詮釋為主的量子力學的輔助說明。基本的物理原理及分析方法在本論文有詳細的說明。
　　基於量子流體力學理論，我們針對量子勢能項，給予更多物理的解釋，量子勢能由於量子擴散的物理現象，可推導出其包含Dilatation energy和Kinetic energy，這個現象統稱為量子擴散流體力學。量子擴散流體力學有以下幾個特點：(1) 量子勢能中由於量子的擴散包含Dilatation energy 和Kinetic energy。(2) 量子擴散流體力學幫助我們更瞭解量子系統中，量子能的變化及平衡等物理現象，這是一般量子流體力學無法辦到。(3) 量子擴散流體力學的優點是提供了一個現象學的解說，但在計算的處理上和一般的量子流體力學是一樣的。

　　本文針對奈米噴嘴的問題，同時提供了數學分析和數值模擬研究。除了數學上的分析以外，為了更為廣泛的應用，如複直的邊界及外加的勢能，所以必須得建立數值模擬的計算能力。在發展計算流體力學之中，有二個問題是必須先克服，第一就是當機率密度很小時，量子勢能項會變的很大，致使計算的模擬中發散。第二就是隨計算時間的迭代，量子波包的傳遞會不會保持大小的一致性，也是很重要的問題。解決了這二個問題之後接下來就是建立兩個噴嘴相互作用的流場模擬。模擬噴嘴的數值方法是二維的量子擴散流體方程式，有限體積法採用三階的MOC上風及限制函數來處理程式中的對流項並利用二階中央差分法處理量子勢能項。在程式驗證上，本論文利用一維和二維的粒子受Echart barrier 和 downhill ramp barrier 二種外加的勢能進行流場模擬並和薛丁格方程式的數值模擬結果互相比對來檢查本程式的正確性，結果顯示本程式所模擬的流場和薛丁格方程式模擬出的流場一致。
　　本文利用此方法來進行二維兩個噴嘴(狹縫)流場的模擬。在模擬的過程中，我們一樣先求得直接解二維噴嘴薛丁格方程式的流場模擬作為發展量子擴散流體計算流場的比較參考。在此分別解出奈米噴嘴之穩態和非穩態流場的數值模擬。最後解出量子擴散流體力學的奈米噴嘴流場。
在分析上，雙孔噴嘴的結構和動態流場，研究上涵蓋的量子雷諾數很廣。
　　研究發現從低的量子雷諾數到小於100中高量子雷諾數，粒子放射出的德布洛依波大，所以量子噴嘴流場結構波的特性很大。當量子雷諾數增大，分支的發生會隨著遠離噴嘴的出口。當量子雷諾數大於400以上，分支的現象已經離噴嘴出口很遠了，同時分支夾角也很小。最後，當量子雷諾數非常大時，流場就變成二個互不相干的噴流結構這和古典的流體力學是相當接近。 另一個值得注意的地方是，當量子雷諾數大於20時，會出現量子叢集的現象。量子叢集現象就是在噴嘴的出口處，會出現一個機率密度很高的區域。其發生的位置也和分支的結構一樣，隨著量子雷諾數的增加而遠離出口處，而且其大小和密度變化也隨著雷諾數的增加而增加，最後像古典粒子的行為一樣。
　　本論文的雙孔量子噴嘴由薛丁格方程式及量子擴散流體力學所計算的數值模擬結果和解析解的流場結構互相比較，整體而言非常的接近。
　　最後奈米噴嘴在現代化的工業是一個相當重要的元件而大量被使用，例如：奈米電路、奈米推進系統、微小型引擎燃油系統、原子干涉儀、蝕刻技術、給藥系統、質量光譜儀和量子感測器等。本論文的目的在於提供一個基礎理論用來 (1) 了解當奈米噴嘴在不同的操作環境下基礎的物理現象；(2) 作為未來不同應用領域設計及最佳化的工具。

　　The focus of this thesis is the theoretical study of the structure and energetic dynamics of dual interacting beams of Fermions emanating from nano-scaled double slit/nozzle configuration, reminiscent the classical double slit experiment, formulated in terms of quantum fluid dynamics (QFD). Particular emphasis is placed on the complementary nature of this theory with the canonical Copenhagen interpretation of quantum mechanics. Basic physical principles and analytical methods are presented in detail.
　　We have investigated this problem both analytically and numerically via a finite difference approach traditionally used in computational fluid dynamics. In addition, we have extended the QFD model to include diffusion. The inclusion of this term gives rise to a “quantum potential,” which has proven to add significant physical insight into fundamental quantum processes, such as the origin of quantum energy. The new approach is referred to as quantum diffusive fluid dynamics or QDFD. QFD and QDFD all have quantum potential. QDFD gives specific views that (1) quantum potential can be related with quantum diffusion induced kinetic energy and dilatation. (2) The QDFD helps us to identify which quantum potential gives rise quantum energy i.e. quantized energy, and which serves to maintain system dynamic equilibria. This would be difficult by QFD. (3) Basically QDFD offers ontological advantage but no computational advantages over QFD.
　　For further applications, complex boundary conditions or external potential, we preformed a finite difference computation on the QDFD equations. Two problems need to be solved when a direct numerical solution of the QDFD. The first is when the probability density is very small and near zero, the quantum potential term becomes singular and causes a divergence in the numerical simulation. The Second is the unitarity in the time evolution of quantum wave packet is significantly. Accurate numerical evaluations are essential to the study of the flow fields created by the interaction of the two jet streams. The numerical implementation of QDFD equation is carried out within the Eularian approach. A third - order, modified Osher-Chakravarthy (MOC) upwind finite-volume scheme was constructed for the convective terms, and a second-order central finite - volume scheme was used to map the field of the quantum potential. As a check on our QDFD finite difference numerical computations we solved the one- and two-dimensional particle motion for Echart barrier and a downhill ramp barrier, respectively. The results were compared to the solution of the Schrödinger equation, using these same potentials, which was solved by a finite difference method.
　　Next we directed our attention to the numerical solution of the nanojet via a finite difference approach. We first solved both the time-independent and time-dependent Schrödinger equation for the two-dimensional nanojet. Last we solved the probability density flow field in QDFD formulation.
　　Analytically, the structural and dynamical behavior of dual nanojets were investigated over a large range of quantum Reynolds numbers. It was found for low to intermediate quantum Reynolds numbers, less than 100, where the deBroglie wavelengths of the emitted particles are large, the dual beam quantum nanojets exhibit a strong wave-like character. This wave-like character manifests itself as a merging of the two separate jet streams into a multiple branched beam with each branch fanning out at different angular displacements relative to the jet axis. As the Reynolds number increases, the merging and branching occur further away from the jet nozzles. At higher Reynolds numbers, 400 and above, branching occurs at a considerable distance from the jet nozzles and the fanning tends to collapse into a seemingly forward propagating beam with little angular spread. Eventually, at very high Reynolds numbers the beam splits into two distinct forward propagating beams with a structure very much like that produced by classical jets. Another remarkably interesting and related feature, which arises for quantum Reynolds numbers ≥ 20, is the formation of quantum clustering. Quantum clustering is the formation of regions in the jet with abnormally high probability density and is the combined result of self-interference within each branch of the jet and quantum tunneling between branches. As with branching, the clusters shift down stream with higher Reynolds numbers. Moreover, the size and density of the cluster also increases with an increase in Reynolds number, again indicating that a classical particle limit is being approached. Branching and clustering are inherent properties of dual beam quantum nanojets and are manifestly brought about by quantum tunneling and interference.
The results of the numerical computations based on both the Schrödinger equation and QDFD are given and compared to those obtained from the analytical solution. Good agreement among the three approaches has been found.
　　Finally, nanojets have a myriad of important, modern day applications for which our model can be used: nano- circuitry and electronics; nano- propulsion / thruster systems; fuel injectors for microscopic engines; atomic optics and interferometry; etching and lithography; drug delivery systems; injecting genes into cells; mass spectrometer; and quantum sensors. Our goal with the present research is to help provide a theoretical foundation to: 1. understand the fundamental physics which governs nanojet operation under various circumstances; and 2. act as tool in guiding the design and optimizing the operational functions for the various applications.

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