量子井，量子線及量子點的數值模擬

Abstract

在這一篇論文裡，我們主要著重於討論多變的量子井、量子線以及量子點並且針對薛丁格方程式運用數值方法去計算其能階及波函數。其中矩陣是在直角座標下運用均勻網格點離散出方程式所得到的。針對組成矩陣所相關的主要想法是以物件導向(C++)的資料結構做為基礎，它能夠被運用在一維，二維及三維模型等的問題上。我們在一維及二維模型的介面條件下考慮一次及二次的逼近方法。 在三維裡，我們專注於建造不同的幾何模型（對稱及非對稱）並且運用二次介面條件去逼近計算他們的最低能階及相關的波函數。而這個資料結構是設計針對在未來發展出更多複雜的模型而能夠輕易的做修改。對於多變的一維、二維及三維所做模擬而的結果也已經得到驗證。

In this thesis, we focus on discussion about various quantum wells, wires and dots using the numerical technique to calculate the energy state and wave function in Schr$\overset{\text{..}}{\text{o}}$dinger's equation. The matrix discretizing the equation uses uniform meshes with rectangular coordinates. The main idea of forming the matrix is based on object-oriented (C++) data structure that can be used for all 1D, 2D, and 3D model problems. We consider linear and quadratic finite difference approximations to the interface condition for 1D and 2D models. In 3D domains, we pay attention to construct different geometrical (symmetric or non symmetric) models to calculate their ground state energies and the corresponding wave function for which the quadratic interface approximation is used. The data structure is designed for easy modifications of more complex models in future development. A great variety of 1D, 2D, and 3D simulation results are demonstrated.