Constants of Motion for Several One-Dimensional Systems and Problems Associated with Getting Their Hamiltonians

The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation; a no-relativistic particle with a time explicitly depending force; a no-relativistic particle with a constant force and time depending mass; and a relativistic particle under a conservative force with position depending mass. The Hamiltonian for these systems, which is determined by getting the velocity as a function of position and generalized linear momentum, can be found explicitly at first approximation for the first system. The Hamiltonians for the other systems are kept implicitly in their expressions for their constants of motion.     

A Hamiltonian Model for Linear Friction in a Homogeneous Medium

 We introduce and study rigorously a Hamiltonian model of a classical particle moving through a homogeneous dissipative medium at zero temperature in such a way that it experiences an effective linear friction force proportional to its velocity (at small speeds). The medium consists at each point in a space of a vibration field modelling an obstacle with which the particle exchanges energy and momentum in such a way that total energy and momentum are conserved. We show that in the presence of a constant (not too large) external force, the particle reaches an asymptotic velocity proportional to this force. In a potential well, on the other hand, the particle comes exponentially fast to rest in the bottom of the well. The exponential rate is in both cases an explicit function of the model parameters and independent of the potential. Received: 18 July 2001 / Accepted: 20 April 2002 Published online: 12 August 2002     

Quantization and instability of the damped harmonic oscillator subject to a time-dependent force

We consider the one-dimensional motion of a particle immersed in a potential field U(x) under the influence of a frictional (dissipative) force linear in velocity () and a time-dependent external force (K(t)). The dissipative system subject to these forces is discussed by introducing the extended Bateman’s system, which is described by the Lagrangian: which leads to the familiar classical equations of motion for the dissipative (open) system. The equation for a variable y is the time-reversed of the x motion. We discuss the extended Bateman dual Lagrangian and Hamiltonian by setting specifically for a dual extended damped–amplified harmonic oscillator subject to the time-dependent external force. We show the method of quantizing such dissipative systems, namely the canonical quantization of the extended Bateman’s Hamiltonian ?. The Heisenberg equations of motion utilizing the quantized Hamiltonian surely lead to the equations of motion for the dissipative dynamical quantum systems, which are the quantum analog of the corresponding classical systems. To discuss the stability of the quantum dissipative system due to the influence of an external force K(t) and the dissipative force, we derived a formula for transition amplitudes of the dissipative system with the help of the perturbation analysis. The formula is specifically applied for a damped–amplified harmonic oscillator subject to the impulsive force. This formula is used to study the influence of dissipation such as the instability due to the dissipative force and/or the applied impulsive force.     

Lie algebraic approach to the time-dependent quantum general harmonic oscillator and the bi-dimensional charged particle in time-dependent electromagnetic fields

We discuss the one-dimensional, time-dependent general quadratic Hamiltonian and the bi-dimensional charged particle in time-dependent electromagnetic fields through the Lie algebraic approach. Such method consists in finding a set of generators that form a closed Lie algebra in terms of which it is possible to express a quantum Hamiltonian and therefore the evolution operator. The evolution operator is then the starting point to obtain the propagator as well as the explicit form of the Heisenberg picture position and momentum operators. First, the set of generators forming a closed Lie algebra is identified for the general quadratic Hamiltonian. This algebra is later extended to study the Hamiltonian of a charged particle in electromagnetic fields exploiting the similarities between the terms of these two Hamiltonians. These results are applied to the solution of five different examples: the linear potential which is used to introduce the Lie algebraic method, a radio frequency ion trap, a Kanai–Caldirola-like forced harmonic oscillator, a charged particle in a time dependent magnetic field, and a charged particle in constant magnetic field and oscillating electric field. In particular we present exact analytical expressions that are fitting for the study of a rotating quadrupole field ion trap and magneto-transport in two-dimensional semiconductor heterostructures illuminated by microwave radiation. In these examples we show that this powerful method is suitable to treat quadratic Hamiltonians with time dependent coefficients quite efficiently yielding closed analytical expressions for the propagator and the Heisenberg picture position and momentum operators.     

Restricted Constant of Motion for the One-Dimensional Harmonic Oscillator with Quadratic Dissipation and Some Consequences in Statistic and Quantum Mechanics

A restricted constant of motion, Lagrangian and Hamiltonian, for the harmonic oscillator with quadratic dissipation is deduced. The restriction comes from the consideration of the constant of motion for the velocity of the particle either for v 0 or for v < 0. A study is done about the implications that these restricted variables have on the specific heat of a thermodynamical system of oscillators with this dissipation, and on the quantization of this dissipative system.     

Dynamics of a vertical flight in the stationary gravitational field of a celestial body: Post-newtonian corrections and gravitational redshift

The standard problem of a radial motion of test particles in the stationary gravitational field of a spherically symmetric celestial body is solved and is used to determine the time features of this motion. The problem is solved for the equations of motion of general relativity (GR), and the time features are obtained in the post-Newtonian approximation, with linear GR corrections proportional to r _g /r and β ² (in the solution being considered, they are of the same order of smallness) being taken rigorously into account. Total times obtained by integrating the time differentials along the trajectories of motion are considered as the time features in question. It is shown that, for any parameters of the motion, the proper time (which corresponds to watches comoving with a test particle) exceeds the time of watches at rest (watches at the surface of the celestial body being considered). The mass and the radius of the celestial body, as well as the initial velocity of the test particle, serve as arbitrary parameters of the motion. The time difference indicated above implies a leading role of the gravitational redshift, which decreases somewhat because of the opposite effect of the Doppler shift. The results are estimated quantitatively for the important (from the experimental point of view) case of vertical flights of rockets starting from the Earth’s surface. In this case, the GR corrections, albeit being extremely small (a few microseconds for several hours of the flight), aremeasurable with atomic (quantum) watches.     

Wigner elementary particle in an external homogeneous field

A covariant Hamiltonian is proposed which permits to describe in the Heisenberg picture the motion of a Wigner elementary particle in a homogeneous electromagnetic field. More precisely, at any time, the elementary particle is in a state associated with a given irreducible representation of the Poincaré group. As a remarkable result, the spin motion is shown to be governed by the Thomas-Bargmann-Michel-Telegdi equation. Also the Galilean limit is discussed.     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

RbNiF3自旋波量子对的Raman散射谱理论分析

在温度由零度直到Néel温度的范围内讨论了RbNiF₃自旋波量子对的Raman散射谱。由一般的对称性原理及实验结果的分析导出磁性系统与辐射场相互作用哈密顿量的具体形式;应用推迟Green函数运动方程方法,得到散射截面的解析表达式;在自旋波量子对传播子中同时计入能量重正化和寿命的效应。最后,在适当的近似下用计算机作数值计算,在没有可调整参数的情况下得到了在Néel温度以下的Raman峰值位置和半宽度随温度的变化关系及各种温度下的Raman谱形,它们与实验结果符合得较好。 关键词：     

Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. I. Allgemeine Theorie und Einschaltvorgang

The general solution of the equations of motion for a charged particle in a magnetic field is given for the following case: the spatially homogeneous magnetic field having a constant direction is a superposition of a field constant in time and one decreasing exponentially in time; taken into account is the influence of the electric field induced by the time dependent magnetic field and a friction force proportional to the particle velocity. The higher transcendental functions appearing in the exact solution are approximated in various ways in dependence on the values of the argument and parameters. The important case of a switching process without a friction force is investigated in detail. The higher transcendential functions can be approximated by simplier functions in such a way, that the solutions for the switching process, valid for all times, differ from the solutions in the case of a linear increasing magnetic field only by factors consisting of elementary functions. Approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The particle orbits are classified and their dependence on the initial values and parameters of the magnetic fields is studied. A comparison between our results and a rectangular variation of the field shows that the latter is not a good approximation for a really exponential increasing field. Finally a detailed investigation shows that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

Contribution to the expression of particle movement in magnetic mirror systems in cyclic variables

The aim of this paper is to express the Hamiltonian function of a particle in a static, axially symmetric magnetic field in convenient variables. Assuming adiabaticity (the relative change of the magnetic field during one cyclotron revolution can be neglected), the Hamiltonian function is determined with cyclicity in two degrees of freedom.For the determination of the Hamiltonian function either the Carthesian, or the orthogonal curvilinear system was used as a starting coordinate system. The latter consists of a natural system of lines of force and equipotentials of the field. In both cases a Hamiltonian function of the formH=H(P₁ P ₂,P ₃,Q ₃) is obtained, where P_i are generalized impulses and Q₃ longitudinal coordinates.The form of the Hamiltonian function is very simple; it facilitates appreciably the integration of the equations of motion and provides simple expressions for the integrals of motion.The author wishes to express his thanks to Ing. J. Váa, the Director of the Institute, for his interest in the work and to Dr. M. Scidl for encouragement.     

One-Dimensional Relativistic Dissipative System with Constant Force and its Quantization

For a relativistic particle under a constant force and a linear velocity dissipation force, a constant of motion is found. Problems are shown for getting the Hamiltonian of this system. Thus, the quantization of this system is carried out through the constant of motion and using the quantization on the velocity variable. The dissipative relativistic quantum bouncer is outlined within this quantization approach. PACS: 03.30.+p, 03.65.−w, 45.05.+x, 45.20Jj     

Tagged Particle Diffusion in One-Dimensional Gas with Hamiltonian Dynamics

We consider a one-dimensional gas of hard point particles in a finite box that are in thermal equilibrium and evolving under Hamiltonian dynamics. Tagged particle correlation functions of the middle particle are studied. For the special case where all particles have the same mass, we obtain analytic results for the velocity auto-correlation function in the short time diffusive regime and the long time approach to the saturation value when finite-size effects become relevant. In the case where the masses are unequal, numerical simulations indicate sub-diffusive behaviour with mean square displacement of the tagged particle growing as t/ln(t) with time t. Also various correlation functions, involving the velocity and position of the tagged particle, show damped oscillations at long times that are absent for the equal mass case.     

Zur Theorie der Bewegung geladener Teilchen in richtungskonstanten Magnetfeldern exponentieller Zeitabhängigkeit. III. Allgemeine Schaltprozesse

The motion of a charged particle in a magnetic field is calculated for the following case: the spatially homogeneous magnetic field having a constant direction is a superposition of a field constant in time and one decreasing exponentially in time; taken into account is the influence of the electric field induced by the time dependent magnetic field and a friction force proportional to the particle velocity. The higher transcendental functions appearing in the exact solutions are approximated in various ways in dependence on the values of argument and parameters. In this manner approximated formulae of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle, and the domain of validity of these formulae is determined. The particle orbits are classified, and their dependence on the initial values, parameters of the magnetic field and on the magnitude of the friction force is studied. A comparison between our results and a rectangular variation of the field is given. It is shown that the electric field induced by the time dependent magnetic field has an important influence on the particle motion.     

Deriving Energy-Gap of Some Hamiltonians with Kinetic Coupling by the Invariant Eigen-Operator Method

We begin with proposing a unitary operator responsible for diagonalizing the Hamiltonian with kinetic couplings in particle physics to get a new form of Hamiltonian which has no coupling terms. By virtue of the invariant eigen-operator (IEO) method we search for the invariant eigen-operators for the new Hamiltonian. In this way the energy-gap of the Hamiltonians can be naturally obtained. This method may be generalized to N-mode Hamiltonian with kinetic couplings case. Work supported by the National Natural Science Foundation of China under grant 10475056 and Foundation of President of Chinese Academy of Science.     

Stochastic rolling of a rigid sphere in weak adhesive contact with a soft substrate

We study the rolling motion of a small solid sphere on a fibrillated rubber substrate in an external field in the presence of a Gaussian noise. From the nature of the drift and the evolution of the displacement fluctuation of the ball, it is evident that the rolling is controlled by a complex non-linear friction at a low velocity and a low noise strength (K), but by a linear kinematic friction at a high velocity and a high noise strength. This transition from a non-linear to a linear friction control of motion can be discerned from another experiment in which the ball is subjected to a periodic asymmetric vibration in conjunction with a random noise. Here, as opposed to that of a fixed external force, the rolling velocity decreases with the strength of the noise suggesting a progressive fluidization of the interface. A state (K) and rate (V) dependent friction model is able to explain both the evolution of the displacement fluctuation as well as the sigmoidal variation of the drift velocity with K. This research sets the stage for studying friction in a new way, in which it is submitted to a noise and then its dynamic response is studied using the tools of statistical mechanics. Although more works would be needed for a fuller realization of the above-stated goal, this approach has the potential to complement direct measurements of friction over several decades of velocities and other state variables. It is striking that the non-Gaussian displacement statistics as observed with the stochastic rolling is similar to that of a colloidal particle undergoing Brownian motion in contact with a soft microtubule.     

Theorie der Bewegung geladener Teilchen in richtungskonstanten zeitabhängigen Magnetfeldern und elektrischen Zusatzfeldern

The motion of a charged particle in spatially homogeneous electric and magnetic fields is calculated for the case of the magnetic field to have a constant direction and its intensity to vary with an arbitrary power of time. The special case of a linearly increasing magnetic field is treated in detail taking into account a friction force proportional to the particle velocity. Generally, the equations of motion are reduced to a single differential equation of second order which is integrated exactly. The higher transcendental functions appearing in the solution are then approximated by elementary functions. Thus asymptotic approximative equations of a very simple form are obtained for position, velocity, kinetic energy and magnetic moment of the particle. The dependence of the particle orbit on the initial values of position and velocity and on the properties of the magnetic field is studied, and it is shown, how the particle motion is a helical motion superposed by a drift. The influence of the electric field induced by the time dependent magnetic field on the particle motion is considered in detail. For an additional electric field being present a drift formula is derived which is a generalization of the well-known ?? × ?? 93 drift for constant fields.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

Rosette motion and string scattering of 20 MeV electrons in MgO single crystals

Analytical estimates and computer simulations were undertaken to perceive the motion of negative particles through a lattice structure, the interaction being classical binary scattering. Three distinct modes of particle motion along atomic strings were found depending on the magnitude of the transverse energy and the angular momentum L of the particle with regard to the string axis. At small and large L increased scattering on the strings as compared with random penetration dominates. At medium L and negative transverse energy (bound state particles in the attractive potential) a rosette motion along the string occurs. In this case small impact parameters to the string atoms are avoided and thus an increased penetrability of the negative particles results. The influence of thermal lattice vibrations on these motions was studied.

Experimentally, the negative particle motion modes manifested themselves in the penetration profiles of 20 MeV electrons through an 8 μm MgO single crystal.