本論文主要研究目的為利用時空守恆法(CESE method)模擬線性與含勢能項之線性薛丁格方程式，其中以線性薛丁格方程式探討自由粒子之行為，和含勢能項之線性薛丁格方程式探討波函數於位能井內之分佈。薛丁格方程式主要是描述粒子的機率分佈，所謂的自由粒子是指粒子在運動過程中不受外力的影響，而含有勢能項的薛丁格方程式，是在探討當粒子碰撞勢能井邊界後所產生的機率分佈情形，其中我們依量子力學的觀點可以知道，在與勢能井碰撞的過程中，粒子是存在穿透與反射兩種機率，而主要影響因素為粒子的總能(E)和位能井(V)的大小，當E>V時，粒子存在穿透與反射，而當E V時，粒子會近似於全反射。
時空守恆法(CESE method)以物理守恆的觀點出發，利用統御方程式滿足通量守恆進行離散，為時間和空間上均具有二階準確度的嶄新數值方法。以時空守恆法數值解與自由粒子之解析解比較可知，時空守恆法模擬線性的薛丁格方程式的機率分佈時，仍具有相當精確之結果；在模擬含有勢能項的線性薛丁格方程式中，當粒子與勢能井碰撞後仍可以得到合理的數值解。此外，研究中還以von Neumann穩定性分析探討時空守恆法在解含勢能項之線性薛丁格方程式時的穩定性。

The main purpose of this research is to utilize the space-time Conservation Element and Solution Element (CESE) method to simulate the linear Schrödinger equation. Where the linear Schrödinger equation describes a free particle behavior, and the Schrödinger equation models the distribution of the wave function within a potential barrier. Schrödinger equation mainly explains the probability distribution of the particle. A free particle motion obeys the Schrödinger equation without external force. Moreover, the Schrödinger equation with potential function can calculate the probability distribution after the wave function collides with the potential barrier. In view of quantum mechanics, we know that in the course of colliding with potential barrier, the particle presents tunneled probability and reflected probability. The main influence factors are the total energy (E) of the particle and the potential barrier (V). When E>V, the particle presents tunnel and reflection; when E V, the particle exhibits completely reflection.
CESE method follows the principle of physical conservation, and derives the discrete formulation of governing equation by means of flux conservation. In time and space, the CESE method is a new numerical scheme with second order accuracy. Comparing the CESE results with the exact solution on the free particle behavior, we demonstrate know that the CESE method can still maintain an accurate result even when the simulation of the probability distribution has a dramatic change. In the simulation of the Schrödinger equation within the potential barrier, the CESE method still received reasonable results after the collision finished. Furthermore, by using the von Neumann stability analysis, this study presents the restriction of numerical parameters of the CESE method for solving the Schrödinger equation.

[1] N. Watanabe, M. Tsukada, “Finite Element Approach for Simulating Quantum Electron Dynamics in a Magnetic Field”,J. Phys. Soc. Japan, 69, 2962-2968, 2000.
[2] J. M. Sanz-Serna, J. G. Verwer, “Conservative and Nonconservative Scheme for the Solution of Nonlinear Schrödinger Equation”, IMA J. Numer. Anal., 6, 25-42, 1986.
[3] S.Jerez, J.V.Romero ,M.D.Rosello ̀,“A Semi-implicit Space-time CE-SE Method to Improve Mass Conservation Through Tapered Ducts in Internal Combustion Engine”,Mathematical and Computer Modelling, 40, 941-951, 2004.
[4] P.Tran,“Solving the Time-Dependent Schrödinger Equation: Suppression of Reflection from the Grid Boundary with a Filtered Split-Operator Approach”, Phys. Rev. E, 58, 8049-8051, 1998.
[5] X.Antoine, A.Arnold, C.Besse, M.Ehrhardt and A.Scha ̈dle,“A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations”, Commun. Comput. Phys., 4, 729-796, 2008.
[6] A. S. Sehra, “Finite Element Analysis of the Schrödinger Equation”, Master Thesis， University of Wales Swansea, 2006.
[7] S. C. Chang, “The Method of Space-Time Conservation Element and Solution Element -A New Approach for Solving the Navire-Stokes and Euler Equation”, J. Comput. Phys., 119, 295-342, 1995.

[8] S. C. Chang, X. Y. Wang, W. M. To, “Application of the Space - Time Conservation Element and Solution Element Method to One- Dimensional Convection-Diffusion Problems”, J. Comput. Phys., 165, 189-215, 2000.
[9] R. C. Greenhow, J. A. D. Matthew ,“Continuum Computer Solutions of the Schrödinger Equation ”, Am. J. Phys., 60,655-663,July 1992.
[10] C.A.Weatherford, E.Red, Albert Wynn III,“Solution of the Time-Dependent Schrödinger Equation Using a Basis in Time”, J.Mol. Struct. (Theochem) 592, 47–51, 2002.
[11] 單溥,陳自強,黃棟洲,“量子物理學”, 復漢出版社,220-222,1989年4月.