无限深方势阱本征值和本征态的三种求解方法

Three kinds of methods for solving eigenvalues and eigenfunctions for the infinite square potential well

一维无限深方势阱模型是量子力学理想模型,经典教材中势阱的边界一般取得比较特殊.或关于坐标原点具有对称性,或势阱左边界位于坐标原点.本文首先展示了如何利用3种方法求解一维任意边界无限深方势阱能量本征值和对应的本征态,不同方法得到的结果彼此之间等价,讨论分析了这3种方法的推导结果,然后得到关于一维任意边界无限深方势阱能量本征值和本征态的通式,从中比较容易看出这两个物理量均与阱宽有关,并且本征波函数与边界值有关,最后将一维结果拓展到二维和三维任意边界无限深方势阱情况.

Moving particle in the one-dimensional infinite square potential well is a quantum ideal model.In the traditional textbooks,the boundary condition of the potential well is particularly taken.Either it is symmetric about the coordinate origin point,or the left boundary value of the well is at the origin of the coordinate.First,it is presented how to solve the energy eigenvalues and eigenfunctions for this model with the arbitrary boundary condition by virtue of the three kinds of approaches.Moreover,the results obtained by the different methods are equivalent to each other.Then the derived results are discussed and analyzed.Furthermore,the general formulas for solving the energy eigenvalues and eigenstates for the one-dimensional infinite square potential well with a arbitrary boundary are acquired.It is easy to see that the two physical quantities are dependent on the well width,and the eigenfunctions are related to the boundary value.Finally,the one-dimensional results are extended to two and three-dimensional infinite square potential wells with the arbitrary boundaries.