Exact invariants for a class of three-dimensional time-dependent classical hamiltonians

An exact invariant is derived for three-dimensional Hamiltonian systems of N particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function f(2)(t), embodying a solution of a third-order differential equation whose coefficients depend on the explicitly known trajectories of the particle ensemble. Our result is applied to a one-dimensional time-dependent nonlinear oscillator and to a system of Coulomb interacting particles in a time-dependent quadratic external potential.     

Derivation of a modified paraxial formalism for two-dimensional trajectories in electron and ion sources of non-simple geometries

A modified paraxial formalism has been developed which describes two-dimensional charged particle beam dynamics in electron and ion sources with pointed or needle-type geometries. A Hamiltonian formalism, and a more exact treatment of energy conservation is used to derive the modified paraxial equation for two-dimensional trajectories in systems with axially symmetric prolate-spheroidal beams. Calculations have been done for a gallium field emission liquid metal ion source modeled by a hyperboloid of revolution and planar extractor. Two important conclusions emerge from these calculations: i) The dominant effect of space-charge, for source geometries with small radii of curvature, occurs in the region close to the apex (<0.05 n) and ii) beam divergence has a strong dependence on geometry. This latter effect is a consequence of large two-dimensional field gradients near the apex of sources with needle-type or pointed geometry.This work was supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-8108829     

Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrödinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Hénon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Hénon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Hénon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.     

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.     

Evolution of Arbitrary States under Fock-Darwin Hamiltonian and a Time-Dependent Electric Field

The method of path integral is employed to calculate the time evolution of the eigenstates of a charged particle under the Fock-Darwin(FD) Hamiltonian subjected to a time-dependent electric field in the plane of the system.An exact analytical expression is established for the evolution of the eigenstates.This result then provides a general solution to the time-dependent Schro¨dinger equation.     

General linear response formula for non integrable systems obeying the Vlasov equation

Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states.     

Entropy of classical systems with long-range interactions

We discuss the form of the entropy for classical Hamiltonian systems with long-range interaction using the Vlasov equation which describes the dynamics of a N particle in the limit N-->infinity. The stationary states of the Hamiltonian system are subject to infinite conserved quantities due to the Vlasov dynamics. We show that the stationary states correspond to an extremum of the Boltzmann-Gibbs entropy, and their stability is obtained from the condition that this extremum is a maximum. As a consequence, the entropy is a function of an infinite set of Lagrange multipliers that depend on the initial condition. We also discuss in this context the meaning of ensemble inequivalence and the temperature.     

Granular matter and the time-dependent viscous eikonal equation

Deposition of granular matter under gravity can be described by the well-known two-layer model for a standing and a rolling layer. Matter from sources enters the rolling layer which flows along the gradient of the standing layer and finally enters the standing layer via interaction of the two layers. From this system of two coupled hyperbolic partial differential equations a time-dependent viscous eikonal equation is derived as a limiting case for weak sources, a thin rolling layer and fast convection of the rolling layer. This equation, supplied with boundary conditions, describes the deposition of dry sand from evenly distributed sources onto a flat table with a vertical rim of variable height. The stationary problem can also be seen as an application of the method of vanishing viscosity to the eikonal equation. For certain types of interaction between the two layers the resulting eikonal equation can be transformed into a linear equation. This transformation yields additional insight into the problem.     

利用概率分布计算电子束聚焦状态

世界各国的学者对电子束的聚焦性能已经做了广泛的研究,提出了许多分析电子束聚焦性能的方法,在这些方法的基础上,试图利用概率分布函数来描述电子束的聚焦过程。该方法将电子的发射看做是随机过程,并从轨迹的落点的概率分析了电子束的形状和大小,避免了杂散小峰的影响。     

Differential conservation laws in nonrelativistic quantum mechanics

Differential conservation laws for quantum mechanical operator fields are studied from a) some general point of view and b) to find the particle, momentum and energy conservation law, if the Hamiltonian isnot local as usually assumed. Concerning (b): If the interaction in a many body system is nonlocal, the particle current operator in the continuity equation has to be redefined. Physical properties of the interaction current are discussed. Similar nonlocal effects must be taken care of in the stress and energy current operator. Concerning (a): Besides the mentioned conservation laws there are arbitrary many other ones. In fact, for each arbitrary field a class of corresponding currents exists and vice versa, which together are related by a differential conservation law. Some physical aspects for defining current operators are given and discussed.     

Optics of oblique meridional and nonmeridional rays of a conicoid

Based on the exact expression for the angular eikonal of an axisymmetric conicoid (a surface formed by revolution of a second-order curve, or conic) the expressions for the Gaussian optics of oblique meridional rays are obtained, which differ from the well-known Gullstrand-Young invariants for a sphere by a set of independent arguments and contain explicitly the parameters of an optical system. The collinear properties of an optical surface with oblique meridional beams are studied. The theory of astigmatism is extended to the case of oblique nonmeridional rays, which makes it possible to construct and analyze in the first approximation the focal surfaces of an actual wide beam of rays. The theory is tested using the example of optimization of an aplanatic lens for the case of an off-axis point of an object. Based on the exact expression for the angular eikonal, definitions of integral aberrations are given. It is proposed to use the method developed in optimization of optical systems.     

The eikonal approximation and E(2) invariance

A group theoretic interpretation is given for the eikonal approximation in potential scattering. This is based upon the approximate invariance at high energies under translations and rotations in the transverse scattering plane; that is, symmetry under the group E(2). The Lippman-Schwinger equation is formulated in a set of basis states which transform invariantly under irreducible representations of E(2) and the solution for the eikonal Hamiltonian, together with lowest order (Saxon-Schiff) corrections is obtained within this basis. A formulation of unitarity in the impact-parameter representation, based upon the E(2) invariance is given. The “geometrical” interpretation of this representation, in connection with the eikonal approximation, is made clear by our approach.     

Ionization of Atoms in a Thermal Field

We study the stationary states of a quantum mechanical system describing an atom coupled to black-body radiation at positive temperature. The stationary states of the non-interacting system are given by product states, where the particle is in a bound state corresponding to an eigenvalue of the particle Hamiltonian, and the field is in its equilibrium state. We show that if Fermi's Golden Rule predicts that a stationary state disintegrates after coupling to the radiation field then it is unstable, provided the coupling constant is sufficiently small (depending on the temperature). The result is proven by analyzing the spectrum of the thermal Hamiltonian (Liouvillian) of the system within the framework of W ^*-dynamical systems. A key element of our spectral analysis is the positive commutator method.     

Fast sweeping method for the factored eikonal equation

We develop a fast sweeping method for the factored eikonal equation. By decomposing the solution of a general eikonal equation as the product of two factors: the first factor is the solution to a simple eikonal equation (such as distance) or a previously computed solution to an approximate eikonal equation. The second factor is a necessary modification/correction. Appropriate discretization and a fast sweeping strategy are designed for the equation of the correction part. The key idea is to enforce the causality of the original eikonal equation during the Gauss–Seidel iterations. Using extensive numerical examples we demonstrate that (1) the convergence behavior of the fast sweeping method for the factored eikonal equation is the same as for the original eikonal equation, i.e., the number of iterations for the Gauss–Seidel iterations is independent of the mesh size, (2) the numerical solution from the factored eikonal equation is more accurate than the numerical solution directly computed from the original eikonal equation, especially for point sources.     

正交电磁场中粒子束的质量聚焦及粒子轨迹

正交电磁场中粒子的运动轨迹在拐点处按指数规律变化,使不同质量离子的轨迹有明显区别,由此可创建新的电磁核素分离法。从牛顿运动方程出发,基于单粒子轨道法阐述被分离核素在正交电磁场中的运动规律,重点找出轨迹的拐点。给出了多质量粒子束在正交电磁场中的质量聚焦特性,以Li离子为例,应用MATLAB模拟粒子束的运动轨迹,结果表明正交电磁场中可实现多质量束流的质量分离。总结出粒子轨迹方程的特点,为多质量束流分离结构的工程化实现提供参数。研究成果可应用到质量分离器、质谱分析仪及材料提纯等装置的研制中,同时对于特殊位形电磁场控制多质量束流等相关领域的研究亦有一定的参考价值。     

正交电磁场中粒子束的质量聚焦及粒子轨迹

<正>交电磁场中粒子的运动轨迹在拐点处按指数规律变化,使不同质量离子的轨迹有明显区别,由此可创建新的电磁核素分离法。从牛顿运动方程出发,基于单粒子轨道法阐述被分离核素在正交电磁场中的运动规律,重点找出轨迹的拐点。给出了多质量粒子束在正交电磁场中的质量聚焦特性,以Li离子为例,应用MATLAB模拟粒子束的运动轨迹,结果表明正交电磁场中可实现多质量束流的质量分离。总结出粒子轨迹方程的特点,为多质量束流分离结构的工程化实现提供参数。研究成果可应用到质量分离器、质谱分析仪及材料提纯等装置的研制中,同时对于特殊位形电磁场控制多质量束流等相关领域的研究亦有一定的参考价值。