一维方势阱束缚态能量本征值的MATLAB行列式分析及广义量子数概念

Analysis of Bound State Energy Eigenvalue of One-dimensional Square Potential Well with MATLAB Determinant and Concept of Generalized Quantum Number

在计算一维方势阱束缚态能量本征值时,基于边界连续性条件,可将相关波函数展开为一个以矩阵形式描述的线性方程组。根据能量本征值必须满足该方程组的系数行列式等于零的要求,在能量区间内逐点扫描,即可确定相应的能量本征值。与其他方法如递推法、转移矩阵法相比,该方法不需要花费较多精力进行编程,具有概念简单、使用方便、实用性强等特点。此外,类似于无限深势阱下定义的量子数概念,可以定义一个广义量子数来描述有限深势阱中能量本征值的分布情况。

In the calculation of the bound state energy eigenvalues of a one-dimensional square potential well,related wave functions can be extended into the linear equations described by a matrix form according to the boundary continuity conditions.Since the energy eigenvalues should satisfy the requirement that the coefficient determinant equals to zero,the corresponding energy eigenvalues can be determined by scanning the energy region point by point.Compared with other methods such as iterative methods and transfer matrix methods etc.,this method is simple in programming and concept,convenient in operation and practical.Besides,like the concept of quantum number defined in the case of an infinite deep square potential well,the distribution of the energy eigenvalues in the finite deep square potential well can be described by defining a generalized quantum number.