论文标题
多项式方法的多项式方法
A Polynomial Approach to the Spectrum of Dirac-Weyl Polygonal Billiards
论文作者
论文摘要
在每个入门量子力学课程中，都可以解决平方或矩形中的Schrödinger方程。其他多边形外壳的解决方案仅存在于非常有限的多边形类别中，并且全部基于Lamé在1852年获得的结果。当然，任何外壳都可以通过有限元方法来解决部分微分方程。在本文中，我们提出了一种差异方法，用于近似于最初用于研究板块振动模式的任意凸多边形外壳的低能频谱和波动功能。鉴于有效模型的有效模型描述了两维材料的量子点的最新兴趣，我们将方法扩展到了以多边形形状的量子台球限制的spin-1/2 fermion，具有不同类型的边界条件。在确切的频谱并将其应用于不存在精确解决方案的情况下，我们说明了该方法的收敛性。
The Schrödinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lamé in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac-Weyl equation for a spin-1/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method's convergence in cases where the spectrum in known exactly and apply it to cases where no exact solution exists.
