We consider the finite potential well defined by ,where n is an integer. The Schrdinger equation is recast into dimensionless form; the size of the well is characterized by a single parameter . We use three different mathematical techniques to solve for the energy eigenvalues and wave functions. Analytic solutions are given for the triangular well (n = 1), the parabolic well (n = 2), and the square well (n → ∞). The finite element method is used to obtain accurate numerical results for 1 ⩽ n ⩽ ∞ over a full range of α. Finally, both the energy and wave function are expanded in a Taylor series in order to study the analytic behavior of the ground state energy (as a function of n) as α → 0.

我们认为由定义的有限电势，其中n是整数。薛定inger方程式被重铸为无量纲形式。井的大小由一个参数来表征。我们使用三种不同的数学方法来求解能量特征值和波动函数。给出了三角形阱（n = 1），抛物线阱（n = 2）和正方形阱（n →∞）的解析解。有限元方法被用于获得1个⩽准确数值结果Ñ在全范围的∞⩽ α。最后，能量和波函数都以泰勒级数展开，以研究基态能量（作为n的函数）为α →0的解析行为。