量子图能谱的数值计算

Numerical calculation of energy spectrum of quantum graph

量子图是指一类定义在由许多一维线连接而成的网络上的量子力学问题.本文介绍了量子图基本概念,以及定义在量子图上的定态问题的数学表述.由波函数及其导数在顶点处满足的边界条件导出了决定量子图的能量本征值的久期方程,证明了久期方程的系数行列式在量子图的边数为偶数时是实数,边数为奇数时则是虚数.这一性质使得可以用数值方法近似地求出任何有限能量范围内的所有能级,并以两种典型的量子图为例,分析了它们的能量本征值随某个特定边长度的变化.

Quantum graphs refer to the quantum mechanical problems which are defined on networks formed by some connecting one-dimensional lines. In this paper, some basic concepts as well as the mathematical formulism of the stationary state problem of the quantum graphs are introduced. From the boundary conditions satisfied by the wave functions and their derivatives at vertices, the secular equation determining the energy eigenvalues of a quantum graph is derived. It is proved that the coefficient determinant of the secular equation is real if the number of edges of the quantum graph is even, but is imaginary if the number is odd. This property makes it possible to numerically calculate all the energy levels in a finite energy range. Two typical quantum graphs are taken as examples to analyze the variation of energy eigenvalues with the change of the length of a specific edge.