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．     

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     

量子能谱中的长程关联

从可积系统求迹公式出发,运用Einstein-Brillouin-Keller(EBK)量子化条件,导出了二维无关联振子系统周期轨道作用量量子化条件,由此发现了量子能级与周期轨道之间的对应关系.这种对应关系表明,如果两条能级对应的周期轨道的拓扑相同,这两条能级对回归函数的贡献相干.回归谱中的一个峰是量子能谱中一组与具有相同拓扑的周期轨道相对应的能级之间相干的结果,这一组能级间存在着长程关联.     

量子能谱中的长程关联

从可积系统求迹公式出发,运用Einstein-Brillouin-Keller(EBK)量子化条件,导出了二维无关联振子系统周期轨道作用量量子化条件,由此发现了量子能级与周期轨道之间的对应关系.这种对应关系表明,如果两条能级对应的周期轨道的拓扑相同,这两条能级对回归函数的贡献相干.回归谱中的一个峰是量子能谱中一组与具有相同拓扑的周期轨道相对应的能级之间相干的结果,这一组能级间存在着长程关联.     

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.     

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.     

周期轨道的量子化与量子能谱中的长程关联

用二维可积系统的半经典量子化方案和二维无关联振子系统的量子能级与周期轨道之间的对应关系,讨论了一组量子能级之间具有长程关联的内在机制,在二维无关联振子系统中,发现了具有相同拓扑M(M1,M2)的周期轨道相对应的量子能级之间存在着长程关联,并以二维4次无关联振子系统为例做了具体说明.     

二维可积系统的求迹公式和周期轨道的量子化

从Berry–Tabor求迹公式出发,导出了二维可积系统周期轨道作用量的半经典量子化条件.利用此量子化条件,考虑周期轨道满足的周期条件,得到了二维无关联四次振子系统周期轨道作用量的半经典量子化条件,并给出了半经典能级公式.对能级与周期轨道的对应关系做了分析.     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质,考虑电子以费米能量穿过台球区域,在台球出口和入口处对入射和出射波函数采用基尔霍夫散射.采用微扰论的Dyson方程得到半经典格林函数,并把赝路径半经典近似作微扰展开得到体系的传输矩阵元.比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度,发现在精度允许范围内它们符合的很好.     

开放量子台球体系散射的赝路径半经典近似

本文讨论正方形量子台球的输运性质,考虑电子以费米能量穿过台球区域,在台球出口和入口处对入射和出射波函数采用基尔霍夫散射.采用微扰论的Dyson方程得到半经典格林函数,并把赝路径半经典近似作微扰展开得到体系的传输矩阵元.比较了传输矩阵元的傅立叶变换谱的峰位置与腔内自由电子经典轨道长度,发现在精度允许范围内它们符合的很好.     

Quantum spectra and classical periodic orbit in the cubic billiard

Quantum billiards have attracted much interest in many fields. People have made a lot of researches on the two-dimensional (2D) billiard systems. Contrary to the 2D billiard, due to the complication of its classical periodic orbits, no one has studied the correspondence between the quantum spectra and the classical orbits of the three-dimensional (3D) billiards. Taking the cubic billiard as an example, using the periodic orbit theory, we find the periodic orbit of the cubic billiard and study the correspondence between the quantum spectra and the length of the classical orbits in 3D system. The Fourier transformed spectrum of this system has allowed direct comparison between peaks in such plot and the length of the periodic orbits, which verifies the correctness of the periodic orbit theory. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

复杂势场量子弹球中疤痕态的量子化条件

量子疤痕是波函数在经典不稳定周期轨道周围反常凝聚的一种量子或波动现象.人们对疤痕态的量子化条件进行了大量研究,对深入理解半经典量子化起到了一定的促进作用.之前大部分研究工作主要集中在硬墙量子弹球上,即给定边界形状的无穷深量子势阱系统.本文研究具有光滑复杂势场的二维量子弹球系统,考察疤痕态的量子化条件及其重复出现的规律,得到了与硬墙弹球不一样的结果,对理解这类现象是一个有益的补充.这些结果将有助于理解具有无规长程杂质分布的二维电子系统的态密度谱和输运行为.     

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.     

Semiclassical limit for a truncated Hamiltonian

In the numerical calculation of the eigenenergies of a polynomial Hamiltonian, the majority of the levels depend on the cutoff of the basis used. By analyzing the finite Hamiltonian matrix as corresponding to a classical "Action Billiard" we are able to explain several features of the full spectrum using semiclassical periodic orbit theory. There are a large number of low-period orbits which interfere at the higher energies contained in the billiard. In this range the billiard becomes more regular than the untruncated Hamiltonian, as reflected by the Berry-Robnik level spacing distribution. (c) 1996 American Institute of Physics.     

Test of Trace Formulas for Spectra of Superconducting Microwave Billiards

Experimental tests of various trace formulas, which in general relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart, for spectra of superconducting microwave billiards of varying chaoticity are reviewed by way of examples. For a two-dimensional Bunimovich stadium billiard the application of Gutzwiller's trace formula is shown to yield correctly locations and strengths of the peaks in the Fourier transformed quantum spectrum in terms of the shortest unstable classical periodic orbits. Furthermore, in two-dimensional billiards of the Limaçon family the transition from regular to chaotic dynamics is studied in terms of a recently derived general trace formula by Ullmo, Grinberg and Tomsovic. Finally, some salient features of wave dynamical chaos in a fully chaotic three-dimensional Sinai microwave billiard are discussed. Here the reconstruction of the spectrum is not as straightforward as in the two-dimensional cases and a modified trace formula as suggested by Balian and Duplantier will have eventually to be applied.     

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.     

Bohmian Mechanics,the Quantum-Classical Correspondence and the Classical Limit: The Case of the Square Billiard

Square billiards are quantum systems complying with the dynamical quantum-classical correspondence. Hence an initially localized wavefunction launched along a classical periodic orbit evolves along that orbit, the spreading of the quantum amplitude being controlled by the spread of the corresponding classical statistical distribution. We investigate wavepacket dynamics and compute the corresponding de Broglie-Bohm trajectories in the quantum square billiard. We also determine the trajectories and statistical distribution dynamics for the equivalent classical billiard. Individual Bohmian trajectories follow the streamlines of the probability flow and are generically non-classical. This can also hold even for short times, when the wavepacket is still localized along a classical trajectory. This generic feature of Bohmian trajectories is expected to hold in the classical limit. We further argue that in this context decoherence cannot constitute a viable solution in order to recover classicality.