二维量子系统Stationary States的边界积分方法
Boundary Integral Method for Stationary States of Two-Dimensional Quantum Systems
摘要：
从基本的层次上详细介绍了计算纳米器件和量子台球中量子粒子定态的边界积分方法。根据该方法，设备或台球的域内的波函数被表示为波函数及其沿域边界的法向导数的线积分;相应的能量本征值被获得为Fredholm行列式的根。该方法的数值实现和应用，以确定与圆形和体育场形状的台球的能级统计和演示混沌运动的量子力学特性。讨论了边界积分法的优点和局限性，以及其它算例的处理方法。
英文摘要：
The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wave function and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.
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