圆环弹子球量子谱的衍射效应

Diffractive effect of quantum spectra in the annular billiard

采用开轨道的量子谱函数,对二维圆环弹子球体系进行了量子谱分析,根据内环半径(f=Rin/Rout)的不同取值,分别计算了相应的量子谱函数的傅里叶变换谱.结果表明,量子峰的位置和粒子运动的经典轨道长度符合得很好,半经典的闭合轨道理论给予了很好的解释;但是随着内环半径的减小,尤其在内环的线度和de Broglie波长可比拟时,量子峰的性质发生了本质性的变化,其特征类似于光学中的衍射图样,这正是由于内环的衍射效应所引起的,非常符合具有波动性的Fresnel-Kirchhoff衍射定理.本文的计算为研究量子台球体系的动力学性质和微腔输运问题提供了理论基础,同时也为研究晶体衍射、光谱分析等提供了一种新的理论方法.

We analyze the quantum spectra of the two-dimensional annular billiard system based on the open-orbit quantum spectrum function, and respectively calculate the Fourier-transformed quantum spectra for various values of a parameter （in the case f = Rin/Rout） . The results show that the peak positions of quantum spectrum match with the lengths of classical orbits of particle motion very well , and the semiclassical closed-orbit theory affords a good explanation. While the inner circle radius becomes comparaber with the de Brogile wavelength of the particle, the character of quantum spectrum changes essentially, turning out to be similar with the pattern of optical diffraction, which is a purely wavelike phenomenon and accords with Fresnel-Kirchhoff diffraction theorem. Our treatment provides a basis theory for researching the property of billiard system and microjunction transport, and opens a new way to investigate the diffraction of crystal and spectral analysis.