从量子谱到经典轨道:圆形腔中的弹子球

From Quantum Spectra to Classical Orbits:the Circular Billiards Systems

用半经典的方法来研究粒子经典运动已成为处理某些量子问题必不可少的工具。周期轨道理论已经成为人们研究定态体系的量子谱和所对应粒子经典运动的关系的主要工具。本文应用周期轨道理论,运用量子谱函数,这种量子谱函数的傅利叶变换包含了体系内许多经典轨道的信息。以二维无限深圆环势阱为例,利用它们的能量本征值和本征函数,用态密度公式计算周期轨道的情况下,量子能态密度的傅立叶变换ρ(L)。在ρ(L)2随L变化的函数图像中得到了一系列的峰,量子峰的位置与用经典方法得到的轨道长度符合得很好,这不但证明经典行为和量子行为具有很好的对应,同时也显示这种半经典的方法是连接经典与量子的一座很好的桥梁。

The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform（ρL） of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.