本論文旨在探討半導體量子點的電子能階。在這裡我們使用單一能帶等效質量模型，建立系統的薛丁格方程式，並且寫成矩陣形式。系統的特徵能量和特徵方程可以藉由對角化矩陣而得到。首先我們嘗試由較簡單的的量子井系統著手，由於量子井系統有解析解，因此可以與這裡所用數值解做比較。
接著我們嘗試拓展到量子點系統。首先我們考慮圓錐形量子點基態電子能階，並且分別外加電場、磁場、異質摻雜，了解量子點電子能階受到不同外力下的變化。接著計算較接近真實結構的金字塔形量子點電子能階，並且與圓錐形量子點計算的結果比較，其中金字塔形量子點可以探討xy 方向的不對稱性。
藉由電洞能階的計算，我們可以探討量子點的電子電洞能階位置所決定的躍遷波長，並且由其波函數分佈可以得到躍遷機率相關的訊息。最後分析探討在量子點異質結構中覆蓋不同應力緩衝層所得到的躍遷波長，及基態與第一激發態遷移之能階差。我們模擬的結果可以定性地解釋實驗上量到的數據。

The electronic structure of quantum dots is calculated by the Raleigh-Ritz variation method combined with effective mass method. We discuss the effect of dots size, electric field, magnetic field, impurity and barrier height.Comparison between the theoretical calculation and experimental results,our simulation can explain the experimental results.

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