Quintela M F C Martins, Lopes Dos Santos J M B
Centro de Física das Universidades do Minho e do Porto, CF-UM-UP Departmento de Física e Astronomia, Universidade do Porto, Rua do Campo Alegre 4169-007, Porto, Portugal.
J Phys Condens Matter. 2020 Oct 19;33(3). doi: 10.1088/1361-648X/abbe77.
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 is known exactly, and apply it to cases where no exact solution exists.
在每一门量子力学入门课程中都会求解具有硬壁的正方形或矩形中的薛定谔方程。其他多边形区域的解仅存在于非常有限的一类多边形中，并且都基于拉梅在1852年得到的一个结果。当然，任何区域都可以用偏微分方程的有限元方法来处理。在本文中，我们提出一种变分方法来近似任意凸多边形区域的低能谱和波函数，该方法最初是为研究平板的振动模式而开发的。鉴于最近对二维材料量子点谱的兴趣，这些量子点由无质量电子的有效模型描述，我们将该方法扩展到适用于具有不同类型边界条件的、限制在多边形形状量子台球中的自旋1/2费米子的狄拉克 - 外尔方程。我们在谱已知精确解的情况下说明了该方法的收敛性，并将其应用于不存在精确解的情况。
