氢原子相干态

由Kustannheimo-Stiefel变换,可将量子力学中的氢原子问题化为带有约束条件的四维各向同性谐振子。在此基础上定义相干态,并证明力学量坐标和动量对相干态的平均,给出经典开普勒运动轨道。同时也讨论该相干态中的测不准关系式。 关键词：     

Quasiclassical dynamics of resonantly driven Rydberg states

We present a semiclassical analysis of the dynamics of Rydberg states of atomic hydrogen driven by a resonant microwave field of linear polarization. The semiclassical quasienergies of the atom in the field are found to be in very good agreement with the exact quantum solutions. The ionization rates of individual eigenstates of the atom dressed by the field reflect their quasiclassical dynamics along classical periodic orbits in the near integrable regime, but exhibit a transition to nonspecific rates when global chaos takes over in phase space. We concentrate both on the principal resonance where the unperturbed Kepler frequency is equal to the driving field frequency and on the higher primary resonance The latter case allows for the construction of nondispersive wave packets which propagate along Kepler ellipses of intermediate eccentricity. Received: 23 June 1998 / Accepted: 10 November 1998     

氢原子在相干态下的Kepler轨道

通过把库仑问题转换成线性谐振子,可以建立二维和三维氢原子不扩散的相干态,力学量在这些相干态中的平均值给出物理空间中Kepler轨道的运动。在变换为谐振子的空间中引入的非物理时间变量起了重要作用。     

Nonlinear rotary wave ion plasma

The nonlinearly coupled Vlasov-Maxwell ion-plasma field equations are solved exactly for a transversely uniform subgroup of rotational modes induced by a uniform axial magnetic field. The ion orbits in momentum space are bipolar doubly periodic eigenfunctions of ion proper time, obtained in closed form as the difference between two doubly quasi-periodic Weierstrass zeta functions. The ion orbits in position space are helical-spiral doubly quasi-periodic functions of ion proper time, expressible simply in terms of doubly quasi-periodic Weierstrass sigma functions. The complete ion distributions are flexible functions of six constants of the ion motion: wave-frame ion energy, transverse gyro center, an inner Hamiltonian correlating wave-frame ion momentum with wave-frame axial position, and both first and second axial integration constants. A rotary electromagnetic plane wave propagates along the axial magnetic field with complex cisoidal dependence upon wave-frame axial position. The eigenvalue determination intricately interrelates the wave propagation vector, the wave amplitude, the axial magnetic field, the double periods, and the bipole separation.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|~(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.     

Rigid Particle and its Spin Revisited

The arguments by Pandres that the double valued spherical harmonics provide a basis for the irreducible spinor representation of the three dimensional rotation group are further developed and justified. The usual arguments against the inadmissibility of such functions, concerning hermiticity, orthogonality, behaviour under rotations, etc., are all shown to be related to the unsuitable choice of functions representing the states with opposite projections of angular momentum. By a correct choice of functions and definition of inner product those difficulties do not occur. And yet the orbital angular momentum in the ordinary configuration space can have integer eigenvalues only, for the reason which have roots in the nature of quantum mechanics in such space. The situation is different in the velocity space of the rigid particle, whose action contains a term with the extrinsic curvature.     

Numerical experiments of the planar equal-mass three-body problem: the effects of rotation

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k相似文献   

Finding the Hamiltonian for Cosmological Models in Fourth-Order Gravity Theories Without Resorting to the Ostrogradski or Dirac Formalism

The Hamilton formalism of cosmological models in fourth-order theories of gravity is considered. An approach to constructing the Hamilton function is presented which starts by replacing the second order derivatives of configuration space coordinates by functions depending on these coordinates, its first order derivatives, and additional variables playing the role of configuration space coordinates. This formalism, which does not resort to the Ostrogradski or Dirac formalism, is elucidated and applied to examples. For a special class of Lagrange functions, it is demonstrated that the canonical coordinates of the considered formalism and of the Ostrogradski formalism are related via a canonical transformation. The canonical transformation is a transformation of the configuration space coordinates and a transformation of momentum components induced by the transformation of the configuration space coordinates for a special element of the class of Lagrange functions mentioned. The Wheeler-DeWitt equations belonging to this Lagrange function are related via minisuperspace coordinate transformations.     

Fractional extensions of the classical isotropic oscillator and the Kepler problem

The class of fractional Hamiltonian systems that generalize the classical problem of the two-dimensional (2D) isotropic harmonic oscillator and the Kepler problem is considered. It is shown that, in the 4D space of structural parameters, the 2D isotropic harmonic oscillator can be extended along a line in such a way that the orbits remain closed and oscillations remain isochronous. Likewise, the Kepler problem can be extended along a line in such a way that the orbits remain closed for all finite motions and the third Kepler law remains valid. These curves lie on the 2D surfaces where any dynamical system is characterized by the same rotation number for all finite motions.     

Periodic orbit quantization of the anisotropic Kepler problem

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.     

Structure and reactions of the Li and Be halo nuclei

The phenomenon of s-wave intruder orbits is examined for a range of Li and Be isotopes. Whether these states are mixed or stand alone, they contribute distinctively to the properties of these nuclei. For weakly bound states, they accentuate those features of large sizes, narrow momentum widths and low-lying E1 strengths that are associated with halo nuclei.     

Periodic solutions of symmetric Kepler perturbations and applications

We investigate the existence of several families of symmetric periodic solutions as continuation of circular orbits of the Kepler problem for certain symmetric differentiable perturbations using an appropriate set of Poincaré-Delaunay coordinates which are essential in our approach. More precisely, we try separately two situations in an independent way, namely, when the unperturbed part corresponds to a Kepler problem in inertial cartesian coordinates and when it corresponds to a Kepler problem in rotating coordinates on ?³. Moreover, the characteristic multipliers of the symmetric periodic solutions are characterized. The planar case arises as a particular case. Finally, we apply these results to study the existence and stability of periodic orbits of the Matese-Whitman Hamiltonian and the generalized Størmer model.     

Coherent states for the Kepler–Coulomb problem on a sphere

An algebraic approach to Kepler problem in a curved space is introduced. By using this approach, the creation and annihilation operators associated to this system and their algebra are calculated. These operators satisfy a deformed Weyl–Heisenberg algebra which can be assumed as a deformed su(2)$\mathfrak{s}\mathfrak{u}\left(2\right)$ algebra. By using this fact, the nonlinear coherent states of this system are constructed. The scalar product and Bargmann representation of this family of nonlinear coherent states are constructed. The present contribution shows that these nonlinear coherent states possess some non-classical features which strongly depend on the Kepler coupling constant and space curvature. Depending on the non-classical measures, the smaller the curvature parameter, the more the non-classical features. Moreover, the stronger Kepler constant provides more non-classical features.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

Variations on the Kepler problem

The elliptical orbits resulting from Newtonian gravitation are generated with a multifaceted symmetry, mainly resulting from their conservation of both angular momentum and a vector fixing their orientation in space—the Laplace or Runge-Lenz vector. From the ancient formalisms of celestial mechanics, I show a rather counterintuitive behavior of the classical hydrogen atom, whose orbits respond in a direction perpendicular to a weak externally-applied electric field. I then show how the same results can be obtained more easily and directly from the intrinsic symmetry of the Kepler problem. If the atom is subjected to an oscillating electric field, it enjoys symmetry in the time domain as well, which is manifest by quasi-energy states defined only modulo ħω. Using the Runge-Lenz vector in place of the radius vector leads to an exactly-solvable model Hamiltonian for an atom in an oscillating electric field—embodying one of the few meaningful exact solutions in quantum mechanics, and a member of an even more exclusive set of exact solutions having a time-dependent Hamiltonian. I further show that, as long as the atom suffers no change in principal quantum number, incident radiation will produce harmonic radiation with polarization perpendicular to the incident radiation. This unusual polarization results from the perpendicular response of the wavefunction, and is distinguished from most usual harmonic radiation resulting from a scalar nonlinear susceptibility. Finally, I speculate on how this radiation might be observed.     

Hole states in single-particle knock-out reactions

The hole states created in single-particle knock-out reactions are physical states but have finite lifetime. The hole is moving in a complex potential, the imaginary part of which is related to the width of the state, and the hole energy and wave functions are complex. The influence of these features on the momentum distributions of the hole is studied for the knock-out of the 1s and 1p protons from ¹²C and is found to be rather insignificant. However it is interesting to note that the overlap integral in momentum space shows a significant change.     

相对论性层子模型中的介子结构波函数

本文讨论了旋量的层子-反层子束缚态的Bethe-Salpeter方程(以下简称B-S方程),假定层子反层子之间只有标量耦合的相互作用,这作用为相对论协变的ν势所描述。介子结构波函数按O(4)群展开后保留最低次分波,进行了数值求解,然后计算了计入高次分波的影响。计算了介子的均方半径。计算结果表明在紧束缚态情况下,低次分波大分量近似是合理的、好的近似。作为对比,对松束缚态也进行了讨论。     

Higgs Algebraic Symmetry of Screened System in a Spherical Geometry

The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng[Z.B.Wu and J.Y.Zeng,Phys.Rev.A 62(2000)032509].We find similar properties in the corresponding systems in a spherical space,whose dynamical symmetries are described by Higgs algebra.There exist extended Runge-Lenz vector for screened Coulomb potentials and extended quadruple tensor for screened harmonic oscillators.They,together with angular momentum,constitute the generators of the geometrical symmetry group.Moreover,there exist an infinite number of closed orbits for suitable angular momentum values,and we give the equations of the classical orbits.The eigenenergy spectrum and corresponding eigenstates in these systems are derived.