Spectra and Wavefunctions in a Ray-Splitting Sinai Microwave Billiard and their Semiclassical Interpretation

Experimental results on spectra and wave functions of a ray-splitting microwave billiard are presented. The billiard is formed by a flat rectangular microwave cavity with a quarter-circle insert made of teflon in one of the corners. Using the Gutzwiller trace formula, the contribution of the periodic orbits of the billiard to the density of states are determined. The wave functions, many of them showing scars associated with periodic orbits, are interpreted in terms of the semiclassical Green function.     

可积系统——求迹公式和h逆谱分析(英文)

研究了二维无关联四次振子系统,有理环面上积分 Hamiltonian运动方程给出了系统一系列周期轨道和经典物理量 ,使用半经典近似下的 Berry- Tabor求迹公式,得到了半经典的态密度.应用 Fourier变换分析了每条周期轨道对态密度的贡献,并与量子态密度的 Fourier变换结果比较证实了半经典求迹公式的有效性.Periodic orbits of two dimensional uncoupled quartic oscillator were calculated by inte grating Hamiltonian equations of motion on reasonable tori, and several classical quantities were also computed. Inserting them into Berry Tabor trace formula, a trace, i.e., the semiclassical density of states of the corresponding quantum system, was obtained. Finally, Fourier transform was adopted to verify the contribution of each periodic orbit. Good agreement between the semiclassical ...     

可积系统棗求迹公式和h逆谱分析

研究了二维无关联四次振子系统,有理环面上积分Hamiltonian运动方程给出了系统一系列周期轨道和经典物理量,使用半经典近似下的Berry-Tabor求迹公式,得到了半经典的态密度．应用Fourier变换分析了每条周期轨道对态密度的贡献,并与量子态密度的Fourier变换结果比较证实了半经典求迹公式的有效性．     

Phase space scars and quantum billiards

A semiclassical expression is derived for the spectral Wigner function of ergodic billiards in terms of a sum over contributions from classical periodic orbits. It represents a generalization of a similar formula by Berry, which does not immediately apply to billiard systems. These results are a natural generalization of Gutzwiller's trace formula for the density of states. Our theory clarifies the origin of scars in the eigenfunctions of billiard systems. However, in its present form, it is unable to predict what states will be dominated by individual periodic orbits. Finally, we compare some of the predictions of our theory with numerical results from the stadium. Within the limitations of numerical resolution, we find agreement between the two.     

Escape of photo-detached electron from an annular nanomicrocavity

The escaped probability density of the photo-detached electron in an annular nanomicrocavity shows strong oscillations as a function of the length of the escape orbits. We present a semiclassical theory that describes theses oscillations in terms of bundles of escape orbits. Due to the interference effects of the electron waves travelling along different escaped orbits, oscillatory structures appear in the escaped probability density. In addition, the calculation results suggest that the escaped probability density of the photo-detached electron is not only related to the inner radius of the annular microcavity, but also related to the laser polarization. In order to show the correspondence between the escaped probability density and the detached electron’s escaped orbits clearly, we calculate the Fourier transformed semiclassical wave function and find that the peak positions agree well with the length of the detached electron’s orbits. We hope that our results will be useful in understanding the escape and propagation process of particles through semiconductor microjunctions or ballistic microstructure.     

Casimir interaction among objects immersed in a fermionic environment

Using ensembles of two, three, and four spheres immersed in a fermionic background we evaluate the (integrated) density of states and the Casimir energy. We thus infer that for sufficiently smooth objects, whose various geometric characteristic lengths are larger then the Fermi wave length one can use the simplest semiclassical approximation (the contribution due shortest periodic orbits only) to evaluate the Casimir energy. We also show that the Casimir energy for several objects can be represented fairly accurately as a sum of pairwise Casimir interactions between pairs of objects.     

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.     

Chaotic spectroscopy

The spectra of quantized chaotic billiards from the point of view of scattering theory are discussed. It is shown how the spectral and resonance density functions both fluctuate about a common mean. A semiclassical treatment explains this in terms of classical scattering trajectories and periodic orbits of the Poincare scattering map. It is shown that this formalism provides an alternative derivation and a new interpretation of Gutzwiller's periodic orbits sum for the spectral density. Moreover, it is a convenient starting point for a derivation of a Riemann-Siegel "look alike" expression for the secular equation in terms of periodic orbits of finite length.     

Semiclassical interpretation of the gross-shell structure in deformed nuclei

An analysis of the classical closed orbits in the infinitely deep elongated ellipsoidal well is given with respect to their relevance to the gross-shell structure of the single-particle spectra in deformed nuclei. It is shown that the deformed-shape gross-shells responsible for the ground state deformations, as well as those responsible for the fission intermediate states, find a reasonable explanation in terms of the semiclassical three-dimensional quantization involving the planar orbits in the axis-of-symmetry plane and the simplest three-dimensional orbits which appear at large distortions. The former are found important for the ground state shapes. The results are also compared with the harmonic oscillator model.     

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     

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.     

Level Density of the Hénon-Heiles System Above the Critical Barrier Energy

We discuss the coarse-grained level density of the Hénon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller’s semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we reproduce the quantum-mechanical density of states very accurately. We also present a perturbative calculation of the stabilities of two infinite series of orbits (R_n and L_m), emanating from the shortest librating straight-line orbit (A) in a bifurcation cascade just below the barrier, which at the barrier have two common asymptotic Lyapunov exponents χ_R and χ_L.     

2D chaotic quantum states in superlattices

The semiclassical motion of an electron along the axis of a superlattice may be calculated from the miniband dispersion curve. Under a weak electric field the electron executes Bloch oscillations which confines the motion along the superlattice axis. When a magnetic field, tilted with respect to the superlattice axis, is applied the electron orbits become chaotic. The onset of chaos is characterised by a complex mixed stable-chaotic phase space and an extension of the orbital trajectories along the superlattice axis. This delocalisation of the orbits is also reflected in the quantum eigenstates of the system some of which show well-defined patterns of high probability density whose shapes resemble certain semiclassical orbits. This suggests that the onset of chaos will be manifest in electron transport through a finite superlattice. We also propose that these phenomena may be observable in the motion of trapped ultra-cold atoms in an optically induced superlattice potential and magnetic quadrupole potential whose axis is tilted relative to the superlattice axis.     

强场中NO分子回归谱的长程散射矩阵的理论研究

采用半经典散射矩阵方法研究外磁场中高里德伯态双原子分子在能量范围为77010—77050cm^-1的回归谱.通过引进模型势简化强磁场中NO分子的高里德伯电子的势函数,找出其在核转动量子数分别为N=1,3,5的三个通道中的闭合轨道,重点分析了强磁场中NO分子的长程散射矩阵元实部的傅里叶变换谱与闭合轨道之间的一一对应关系.     

Hadronic density of states from string theory

We present an exact calculation of the finite temperature partition function for the hadronic states corresponding to a Penrose-Güven limit of the Maldacena-Nù?ez embedding of the N=1 super Yang-Mills (SYM) into string theory. It is established that the theory exhibits a Hagedorn density of states. We propose a semiclassical string approximation to the finite temperature partition function for confining gauge theories admitting a supergravity dual, by performing an expansion around classical solutions characterized by temporal windings. This semiclassical approximation reveals a hadronic energy density of states of a Hagedorn type, with the coefficient determined by the gauge theory string tension as expected for confining theories. We argue that our proposal captures primarily information about states of pure N=1 SYM theory, given that this semiclassical approximation does not entail a projection onto states of large U(1) charge.     

Quantum corrections to the semiclassical level density of the circular disk in homogeneous magnetic fields

In the semiclassical trace formula for the level density of a circular disk in homogeneous magnetic fields, quantum corrections to the Maslov phase have been shown to be important in strong fields. In this article further quantum corrections are considered, namely, grazing corrections which are relevant for whispering-gallery orbits, and a uniform approximation to the bifurcation points in strong fields is applied. Both corrections are shown to have a surprisingly small effect on the semiclassical level density. Implementing those corrections requires a technique different from the common Gaussian smoothing in the numerical evaluation of the trace formula. The appropriate generalization is presented. 1997 Elsevier Science B.V. All rights reserved.     

Semiclassical treatment of light-ion induced alignment of theL 3-subshell in heavy atoms

Light ion induced alignment measurements ofL ₃-subshell in heavy atoms are analyzed within the framework of the semiclassical theory of inner shell ionization. Coulomb deflection effects are exactly taken into account by hyperbolic classical orbits while the electron in the initial and final states is approximated by screened relativistic hydrogenic wave functions. Satisfactory agreement between experiments and theory is obtained for proton- and α-particle induced reactions within the whole region of bombarding energies.