The quantum spectra analysis of the circular billiards in wells

We use a recently defined quantum spectral function and apply the method of closed-orbit theory to the 2D circular billiard system. The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in the well. We study the correspondence between quantum spectra and classical orbits in the circular, 1/2 circular and 1/4 circular wells using the analytic and numerical methods. We find that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical orbits. These examples show evidently that semi-classical method provides a bridge between quantum and classical mechanics.     

半圆和四分之一圆二维弹子球的量子谱分析

采用近来提出的量子谱函数,我们把闭合轨道理论应用到半圆和四分之一圆弹子球系统,这种量子谱函数的傅利叶变换包含了连接任意两点的许多经典轨道的信息.计算表明量子谱的傅立叶变换和经典轨道的长度符合的很好.从这两个体系可以看出半经典理论为经典和量子力学提供了很好的桥梁作用.     

圆环弹子球量子谱的衍射效应

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=R_in/R_out）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. 关键词： 圆环弹子球 量子谱函数 傅里叶变换 衍射效应     

二维椭圆量子台球中的谱分析

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 method for stationary states，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. 关键词： 椭圆量子台球 定态展开方法 闭合轨道理论 量子谱     

矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

偏心环形开放台球中粒子逃逸动力学的分形性质

二维台球体系因为能够体现混沌现象的基本特征且数值运算相对简单,从而成为研究微观体系混沌动力学的理想模型,近年来一直广受关注.本文研究非同心的环形开放台球中粒子逃逸的混沌动力学性质,它体现了与初条件密切相关的奇异性.采用简化的盒计数 (box-counting)算法,计算了分形维数,结果定量地反映了粒子逃逸前与环壁碰撞次数随粒子入射角变化的函数关系.其中,特别关注环形台球的偏心率对体系混沌性质的影响.     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

The closed-orbit and the photoabsorption spectra of lithium atom in varying＇magnetic fields

Using a simple analytic formula from closed orbit theory, we have calculated the photoabsorption spectra of Li atom in different magnetic fields. Closed orbits in the corresponding classical system have also been obtained for B=5.96T.We demonstrate schematically that the closed orbits disappear gradually with the decrease of the magnitude of the magnetic field. This gives us a good method to control the closed orbits in the corresponding system by changing the magnetic field, and thus changing the peaks in the photoabaorption spectra. By comparing the photoabsorption spectra of Li atom with those of hydrogen case, we find the core-scattered effects play an important role in multi-electron Rydberg atoms.     

Non-Poissonian statistics in an optical analog of quantum billiard with perfectly square boundaries

We study deviation from the Poissonian statistics of the frequency spacing distribution, appearing due to coupling of polarizational and transverse degrees of freedom in a perfectly square vertical cavity surface emitting laser. The deviation can be controlled by strength of the intracavity anisotropy and its alignment to the device boundaries.     

Trace formula for level density of a spherical billiard

A trace formula for the oscillating part of the level density for a spherical billiard has been obtained in spherical polar coordinates. The Jacobian of stability and the length of the orbits are obtained from the classical mechanics of the problem. The same formula is applicable to both the planar and the diametric orbits. Numerical results have been obtained with this formula and compared with the results from exact quantum theory, EBK quantization, and Balian and Bloch.     

Closed orbits and semiclassical wavefunctions in two-dimensional Hamiltonian systems

The periodic-orbit theory of the density-of-states and the closed-orbit theory of atomic absorption spectra relate properties of a quantum system to properties of periodic or of closed orbits of a system. In these theories, every return of an orbit to the initial point makes a contribution, so in general for each orbit an infinite number of terms must be computed and summed. We show that the term asising from thenth return of an orbit can be calculated from properties of the orbit on its first return.     

Static and time-dependent perturbations of the classical elliptical billiard

The elliptical billiard problem defines a two-dimensional integrable discrete dynamical system. Integrability not being a robust property, we study some static and time-dependent perturbations of this problem. For the static case, we observe the transition from integrability to chaos, on some perturbations of the ellipse. Then we study time-dependent perturbations, supposing that the boundary deforms periodically with the time, remaining always an ellipse. We investigate numerically the now four-dimensional phase space, looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely.     

正方体量子台球的半经典分析(英文)

近几年,量子台球问题引起人们的广泛兴趣.以前有很多对二维量子台球做过研究,相对于二维台球来说,三维台球更接近实际体系.本文以三维正方体量子台球为例,利用半经典闭合轨道理论计算了正方体量子台球中的经典开轨道,并研究量子谱函数与经典轨道长度之间的对应关系,发现他们之间对应的很好.这将有助于我们分析开放型量子台球中输运性质问题.利用这种方法物理图像清晰,计算量小并且可以帮助理解一些混沌体系的性质.这是半经典理论为联系量子力学与经典力学起桥梁作用的又一证明.     

等腰直角三角形的二维量子谱和经典轨道

本文研究了等腰直角三角形中基于其波函数和能级结构的二维量子谱．虽然这个量子台球系统的本征态是无法分离的,但是关于两个变量的问题是完全可解的．通过对二维量子系统的波函数做相应的傅里叶变换,得到了系统的二维量子谱,把得到的结果和经典的二维量子台球轨道做相应性的对比发现：傅里叶变换的量子谱的峰值位置和经典轨道的长度之间存在着很好的对应关系,这说明经典计算的结果和量子计算的结果符合得非常好,从而进一步验证了周期轨道理论的正确性．     

Semiclassical Calculation of Recurrence Spectra of Li Rydberg Atom in Crossed Electric and Magnetic Field

Closed-orbit theory is a semiclassical technique for explaining the spectra of Rydberg atoms in external fields. Using the closed-orbit theory and classical perturbation theory, we calculate the scaled recurrence spectra of Lithium atom in magnetic field plus a weak perpendicular electric field. The results show when the crossed electric field is added, the recurrence spectra are weakened greatly. As the scaled electric field f increases, the peaks of the recurrence spectra lose strength. Some recurrences are very sensitive and fall off rapidly as f increases; others persist till much higher f . As the electric field is stronger, some of the peaks revive. This phenomenon, caused by the interference among the electron waves that return to the nucleus, can be computed from the azimuthal dependence of the classical closed orbits.     

强磁场中Rydberg NO分子的回归谱研究

利用分子的闭合轨道理论和多通道量子亏损理论,考虑核的转动影响,计算了Rydberg NO分子在外磁场中的回归谱.研究表明,复杂谱的结构可以用包含分子实散射的闭合轨道理论来解释,不管是同一通道内的弹性散射还是不同通道间的非弹性散射都起着非常重要的作用,尤其是较大标度作用量下,散射作用远大于基本轨道和重复轨道的贡献,很大程度上影响着体系的动力学性质。     

外电场中Li原子回归谱的分岔效应

半经典闭合轨道在外场和激光参数的某些值附近发生分岔，导致波函数的发散，使原有的半经典闭合轨道理论失效-计算了高激发的Li原子在标度能量为ε=-2-94，标度电场在135 -86-1/4<160-11范围内光吸收谱的Fourier变换，分析了轨道分岔现 象及其影 响，并采用了一种适用于不同能域的统一近似方法修正了分岔点附近波函数-计算中考虑了 原子实多重散射组合回归效应，并把所得回归谱与用标准半经典闭合轨道理论对Li原子及H 原子的计算结果相比较，证明了轨道分岔效应的重要性- 关键词： 分岔 统一近似 半经典闭合轨道理论 回归谱     

Double-Pulse Spectra and Closed-Orbits: Photodetachment of H-in Parallel Electric and Magnetic Fields

We derive a formula for double-pulse spectra from closed-orbit theory. We then calculate the double-pulsephotodetachment spectra of H- in the presence of parallel electric and magnetic fields. We analyze the spectra in termsof closed-orbits of the system. We suggest a method for the measurement of a phase associated with each closed-orbit.