论文标题
通过B-Spline搭配求解Schrödinger's方程
Solving Schrödinger's equation by B-spline collocation
论文作者
论文摘要
自1970年代初以来，B-Spline搭配技术已被应用于Schrödinger's方程，但是该文献中明显缺少的一个方面是使用高斯点（即LegendReredornomials的零）作为搭配点，这可以显着减少近似错误。过去，作者使用了相同间隔或非线性分布的搭配点（注意到后者可以提高近似准确性），但奇怪的是，它继续避免避免高斯搭配点。 Using the methodology and computer routines provided by Carl de Boor's book A Practical Guide to Splines as a `numerical laboratory', the present dissertation examines how the use of Gaussian points can interact with other features such as box size, mesh size and the order of polynomial approximants to affect the accuracy of approximations to Schrödinger's bound state wave functions for the electron in the hydrogen atom.我们探讨了高斯点处的B型搭配是否可以比同等间隔和非线性分布的搭配点产生更准确的近似值。我们还将B-Spline搭配在高斯点应用于具有立方非线性的Schrödinger方程，该方程过去已被广泛用于研究非线性现象。我们的计算机实验表明，高斯点的搭配可能是氢原子的非常成功的方法，始终优于均匀间隔的搭配点，通常优于非线性分布的搭配点。但是，我们确实会遇到某些情况，通常是当网格相对于氢原子的盒子大小而言，以及在CuxicSchrödinger方程式的情况下，非线性分布式搭配点的性能优于高斯搭配点。
B-spline collocation techniques have been applied to Schrödinger's equation since the early 1970s, but one aspect that is noticeably missing from this literature is the use of Gaussian points (i.e., the zeros of Legendre polynomials) as the collocation points, which can significantly reduce approximation errors. Authors in the past have used equally spaced or nonlinearly distributed collocation points (noticing that the latter can increase approximation accuracy) but, strangely, have continued to avoid Gaussian collocation points. Using the methodology and computer routines provided by Carl de Boor's book A Practical Guide to Splines as a `numerical laboratory', the present dissertation examines how the use of Gaussian points can interact with other features such as box size, mesh size and the order of polynomial approximants to affect the accuracy of approximations to Schrödinger's bound state wave functions for the electron in the hydrogen atom. We explore whether or not, and under what circumstances, B-spline collocation at Gaussian points can produce more accurate approximations to Schrödinger's wave functions than equally spaced and nonlinearly distributed collocation points. We also apply B-spline collocation at Gaussian points to a Schrödinger equation with cubic nonlinearity which has been used extensively in the past to study nonlinear phenomena. Our computer experiments show that collocation at Gaussian points can be a highly successful approach for the hydrogen atom, consistently superior to equally spaced collocation points and often superior to nonlinearly distributed collocation points. However, we do encounter some situations, typically when the mesh is quite coarse relative to the box size for the hydrogen atom, and also in the cubic Schrödinger equation case, in which nonlinearly distributed collocation points perform better than Gaussian collocation points.
