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：     

Geometric quantization of the five-dimensional Kepler problem

An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions. Thus a new regularization of the Kepler problem is established. Then, following Dirac, we quantize the extended phase space, imposing constraint conditions as superselection rules. In that way the interchangeability of the reduction and the quantization procedures is proved.     

A self-consistent field for the H2 molecule

We investigate the simple harmonic oscillator in a 1D box, and the 2D isotropic harmonic oscillator problem in a circular cavity with perfectly reflecting boundary conditions. The energy spectrum has been calculated as a function of the self-adjoint extension parameter. For sufficiently negative values of the self-adjoint extension parameter, there are bound states localized at the wall of the box or the cavity that resonate with the standard bound states of the simple harmonic oscillator or the isotropic oscillator. A free particle in a circular cavity has been studied for the sake of comparison. This work represents an application of the recent generalization of the Heisenberg uncertainty relation related to the theory of self-adjoint extensions in a finite volume.     

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。     

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.     

COMPLEX REPRESENTATION OF PLANAR MOTIONS AND CONSERVED QUANTITIES OF THE KEPLER AND HOOKE PROBLEMS

Using a complex representation of planar motions, we show that the dynamical conserved quantities associated to the isotropic harmonic oscillator (Fradkin–Jauch–Hill tensor) and to the Kepler's problem (Laplace–Runge–Lenz vector) find a very simple and natural interpretation. In this frame we also establish in an elementary way the relation which connects them.     

Kepler motion on single-sheet hyperboloid

The classical Kepler–Coulomb problem on the single-sheeted hyperboloid H ³ ₁ is solved in the framework of the Hamilton–Jacobi equation. We have proven that all the bounded orbits are closed and periodic. The paths are ellipses or circles for finite motion.     

On geometric quantization and its connections with the Maslov theory

A modification of the Konstant-Souriau geometric quantization theory is proposed. It includes the case of a multi-odd-dimensional harmonic oscillator which is unquantizable in the Konstant-Souriau theory. The connections between geometric quantization and the Maslov theory of first term in the WKB asymptotic series are considered. Examples of a multidimensional harmonic oscillator, the three-dimensional Kepler problem and a relativistic mass-spin particle are computed.     

Remarks on the geometric quantization of the Kepler problem

The geometric quantization of the (three-dimensional) Kepler problem is readily obtained from the one of the harmonic oscillator using a Segre map. The physical meaning of the latter is discussed.     

Bohr-Sommerfeld conditions in geometric quantization

The corrected Bohr-Sommerfeld quantum conditions, ∫ pdq?d = integer, are studied in the framework of geometric quantization. It is shown, in the representation given by a polarization F, that a half-form corresponds to a wave function only if it vanishes on all closed curves with tangent vectors in F for which the quantum condition is not satisfied. The constant d is determined, for each closed curve y, by the element of the holonomy group of a bundle of metalinear frames for F induced by y. This result is applied to a one-dimensional harmonic oscillator and a two-dimensional relativistic Kepler problem. In the case of the one-dimensional harmonic oscillator there are two possibilities of choosing a metalinear frame bundle for F. One choice leads to the original Bohr-Sommerfeld condition while the other leads to the corrected version with d = $\text{1}\text{2}$. Similarly, choosing different metalinear frame bundles for F, we get for the relativistic Kepler problem the fine structures of the energy levels corresponding to spin 0 and spin $\text{1}\text{2}$.     

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.     

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.     

Quantum damped oscillator II: Bateman’s Hamiltonian vs. 2D parabolic potential barrier

We show that quantum Bateman’s system which arises in the quantization of a damped harmonic oscillator is equivalent to a quantum problem with 2D parabolic potential barrier known also as 2D inverted isotropic oscillator. It turns out that this system displays the family of complex eigenvalues corresponding to the poles of analytical continuation of the resolvent operator to the complex energy plane. It is shown that this representation is more suitable than the hyperbolic one used recently by Blasone and Jizba.     

Features of the moment functions of an oscillator with parametric instability due to dichotomous noise with Erlang distribution functions

We study stability of the periodic solutions of a linear undamped oscillator with frequency modulated by dichotomous noise whose statistic is determined by the Erlang distribution. It is shown that the amplitudes of harmonic oscillations of such an oscillator increase with time at different rates determined by the ratio of the oscillator eigenfrequency to the characteristic frequency of dichotomous noise. To solve the problem, we use a finite system of closed equations with respect to the moment functions, which was obtained without assumption on the quasi-Gaussianity and delta-correlatedness of the studied process.     

经典轨道的封闭性和径向Schroinger方程的因式分解

研究表明 ,保证经典轨道具有封闭性的 Bertrand定理可以进一步推广 ,在适当的角动量下 ,仍存在着非椭圆的闭合轨道 .对于屏蔽 Coulomb场,可获得广义Runge-Lenz矢量.这种轨道封闭性与径向 Schroodinger方程因式分解相对应. It is shown that for a particle with suitable angular momenta in the screened Coulomb potential or isotropic harmonic potential, there still exists closed orbits rather than ellipse, characterized by the conserved perihelion and aphelion vectors, i.e., extended Runge Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry SO 3. For the potential, factorization of the radial Schrdinger equation to produce raising and lowering operators is also pointed out.     

经典轨道的封闭性和径向Schr?dinger方程的因式分解

研究表明,保证经典轨道具有封闭性的Bertrand定理可以进一步推广,在适当的角动量下,仍存在着非椭圆的闭合轨道．对于屏蔽Coulomb场,可获得广义Runge-Lenz矢量．这种轨道封闭性与径向Schr?dinger方程因式分解相对应．     

Dynamical symmetries and Nambu mechanics

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.     

Quantum-classical correspondence for motion on a plane with deficit angle

A particle constrained to move on a cone and bound to its tip by harmonic oscillator and Coulomb-Kepler potentials is considered both in the classical as well as in the quantum formulations. The SU(2) coherent states are formally derived for the former model and used for showing some relations between closed classical orbits and quantum probability densities. Similar relations are shown for the Coulomb-Kepler problem. In both cases a perfect localization of the densities on the classical solutions is obtained even for low values of quantum numbers.     

Semiclassical shell-structure moment of inertia for equilibrium rotation of a simple Fermi system

Semiclassical shell-structure components of the collectivemoment of inertia are derived within the mean-field cranking model in the adiabatic approximation in terms of the free-energy shell corrections through those of a rigid body for the statistically equilibriumrotation of a Fermi system at finite temperature by using the nonperturbative extended Gutzwiller periodic-orbit theory. Their analytical structure in terms of the equatorial and 3-dimensional periodic orbits for the axially symmetric harmonic oscillator potential is in perfect agreement with the quantum results for different critical bifurcation deformations and different temperatures.     

Dynamics on the cone: Closed orbits and superintegrability

The generalization of Bertrand’s theorem to the case of the motion of point particle on the surface of a cone is presented. The superintegrability of such models is discussed. The additional integrals of motion are analysed for the case of Kepler and harmonic oscillator potentials.