二维椭圆量子台球中的谱分析

Spectra analysis of the two-dimensional elliptic billiards

研究了二维椭圆台球中的量子谱和经典轨道之间的对应关系.为尝试求解没有解析波函数和本征能量又不能分离变量的体系,采用了定态展开方法(expansion methodfor stationarystates,简称EMSS)得到尽可能精确的数值解,这是闭合轨道理论被推广到计算开轨道的情况.比较了傅里叶变换谱和经典轨道,发现量子谱的峰位置与经典轨道的长度在可分辨的范围内符合得很好,这是半经典理论为经典与量子力学的联系提供桥梁作用的又一个例子.

The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states （EMSS） to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.