Coherent quantum states of a relativistic particle in an electromagnetic plane wave and a parallel magnetic field

We analyze the solutions of the Klein–Gordon and Dirac equations describing a charged particle in an electromagnetic plane wave combined with a magnetic field parallel to the direction of propagation of the wave. It is shown that the Klein–Gordon equation admits coherent states as solutions, while the corresponding solutions of the Dirac equation are superpositions of coherent and displaced-number states. Particular attention is paid to the resonant case in which the motion of the particle is unbounded.     

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.     

Two-dimensional motion of a parabolically confined charged particle in a perpendicular magnetic field

The classical two-dimensional motion of a parabolically confined charged particle in presence of a perpendicular magnetic is studied. The resulting equations of motion are solved exactly by using a mathematical method which is based on the introduction of complex variables. The two-dimensional motion of a parabolically charged particle in a perpendicular magnetic field is strikingly different from either the two-dimensional cyclotron motion, or the oscillator motion. It is found that the trajectory of a parabolically confined charged particle in a perpendicular magnetic field is closed only for particular values of cyclotron and parabolic confining frequencies that satisfy a given commensurability condition. In these cases, the closed paths of the particle resemble Lissajous figures, though significant differences with them do exist. When such commensurability condition is not satisfied, path of particle is open and motion is no longer periodic. In this case, after a sufficiently long time has elapsed, the open paths of the particle fill a whole annulus, a region lying between two concentric circles of different radii.     

The classical equations of motion for a spinning point particle with charge and magnetic moment

The classical, special relativistic equations of motion are derived for a spinning point particle interacting with the electromagnetic field through its charge and magnetic moment. Radiation reaction is included. The energy tensors for the particle and for the field are developed as well-defined distributions; consequently no infinities appear. The magnitude of spin and the rest mass are conserved.     

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.     

Classical spin theory

Tensor and vector equations of motion of a classical charged particle with spin have been derived within the framework of the special theory of relativity on the basis of Frenkel's tensor. The anomalous magnetic moment of the particle is considered in a natural manner in deriving the equations. The expression for the forces acting on the particle is constructed with consideration of the effect of spin on the motion trajectory. The spin equations proved to coincide with those obtained previously by Nyborg and Good. The properties of these equations have been studied, and it has been shown that the various equations are in fact variant forms of one and the same equation. In the absence of an anomalous magnetic moment the tensor equation coincides with Frenkel's spin equation, and in the same situation the vector equation transforms to the equation obtained by Tamm. In the special case of homogeneous fields the vector equation coincides with the well-known Bargmann-Michel-Telegdi equation. In conclusion we present spin motion equations for a particle with electric and magnetic charges.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 67–76, February, 1980.     

A charged particle in a homogeneous magnetic field accelerated by a time-periodic Aharonov–Bohm flux

We consider a nonrelativistic quantum charged particle moving on a plane under the influence of a uniform magnetic field and driven by a periodically time-dependent Aharonov–Bohm flux. We observe an acceleration effect in the case when the Aharonov–Bohm flux depends on time as a sinusoidal function whose frequency is in resonance with the cyclotron frequency. In particular, the energy of the particle increases linearly for large times. An explicit formula for the acceleration rate is derived with the aid of the quantum averaging method, and then it is checked against a numerical solution and a very good agreement is found.     

On Some Solutions of Relativistic Wave Equations

In this paper we present exact solutions of the Klein-Gordon and the Dirac equations in different configurations of an electromagnetic field, which are characteristic for free-electron laser-type gauges. In the case of motion of a charge scalar particle in standing wave an energy spectrum is studied. For the motion of an electron in a so-called wiggler magnetic field a spinor wave function is proved to be obtainable. An undulator field configuration with propagating wave is treated also.     

Collective excitations of a 1D Bose–Einstein condensate in an anharmonic trap

We investigate the collective excitations of a one-dimension Bose–Einstein condensate trapped in an anharmonic potential by solving the time-dependent Tonks–Girardeau equation. The governing equations of motions for the low-energy excitations are obtained using variational approaches. The motion of a 1D BEC in a harmonic trap is just like the motion of one particle in a harmonic trap. And quartic distortion of the potential causes the blue-shift and red-shift on the excitation frequency while cube distortion only causes the red-shift.     

Motion of a particle in a static magnetic field and in the field of a electromagnetic plane wave

The solutions of the equations of motion of a charged particle in an external electromagnetic field consisting of a superposition of a constant uniform magnetic field and the field of a circularly polarized electromagnetic plane wave are presented as solutions of the Cauchy problem. The resonance case is studied. Zh. Tekh. Fiz. 67, 94–99 (February 1997)     

Quantum mechanical treatment of a constrained particle on two dimensional sphere

In this work, we study the motion of a particle on two dimensional sphere. By writing the Schrodinger equation, we obtain the wave function and energy spectra for three dimensional harmonic oscillator potential plus trigonometric Rosen–Morse non-central potential. By letting three special cases for intertwining operator, we investigate the energy spectra and wave functions for Smorodinsky–Winternitz potential model.     

中性粒子在Ioffe阱中的经典运动方程的Floquet解

研究中性粒子在Ioffe阱中近原点区域的囚禁时,阱中的磁场可以呈现出一种简明的形式.磁矩μ反平行于磁场的中性粒子在阱中与磁场发生相互作用,借助相互作用势,可以获得粒子在阱中的经典运动方程.在一定的条件下,采用迭代近似的目的,将方程演化为马丢方程的形式,利用传统的WKBJ目的可实现方程的近似求解.研究阱中中性粒子的囚禁问题时,感兴趣的是马丢方程的Floquet解,即周期为π,2π的全周期和半周期解,欲获得这种周期解,马丢方程中的参数λ和q必须满足一定的关系,为此必须选择阱的特定参数和粒子的特定初始条件,对这一问题进行了探索性的研究.     

磁场中带电粒子阻尼运动的分析力学表示

研究带电粒子在磁场中作阻尼运动的分析力学表示. 首先, 求解运动微分方程的Birkhoff力学逆问题, 得到带电粒子的4个Rirkhoff表示; 其次, 导出4个状态空间中Lagrange表示和对应的4个位形空间中Lagrange表示; 第三, 构造出4个Hamilton函数; 最后, 从粒子运动的分析力学表示直接得到4个第一积分, 并求出运动方程的解.     

Motion of a charged fermion with an anomalous magnetic moment in magnetized media

A quantization method based on the use of lowering and raising operators is developed and applied to describing states of Fermi particles that move under extreme external conditions (strong magnetic field and dense matter). The efficiency of this method is demonstrated by applying it to examples of finding exact solutions of quantum equations that describe the motion of charged particles in a magnetic field and dense matter. For the first time, the problem of charged-fermion motion in matter and an external magnetic field is formulated and solved with allowance for the anomalous magnetic moment of the particle. Exact solutions for the wave functions and energy spectrum of the respective modified Dirac equation are obtained. The application of these results to describing fermions and neutrinos is of special interest for astrophysical applications.     

Basic features of a charged particle dynamics in a laser beam with static axial magnetic field

In this paper, the trajectory and kinetic energy of a charged particle, subjected to interaction from a laser beam containing an additionally applied external static axial magnetic field, have been analyzed. We give the rigorous analytical solutions of the dynamic equations. The obtained analytical solutions have been verified by performing calculations using the derived solutions and the well known Runge-Kutta procedure for solving original dynamic equations. Both methods gave the same results. The simulation results have been obtained and presented in graphical form using the derived solutions. Apart from the laser beam, we show the results for a maser beam. The obtained analytical solutions enabled us to perform a quantitative illustration, in a graphical form of the impact of many parameters on the shape, dimensions and the motion direction along a trajectory. The kinetic energy of electrons has also been studied and the energy oscillations in time with a period equal to the one of a particle rotation have been found. We show the appearance of, so-called, stationary trajectories (hypocycloid or epicycloid) which are the projections of the real trajectory onto the (x, y) plane. Increase in laser or maser beam intensity results in the increase in particle’s trajectory dimension which was found to be proportional to the amplitude of the electric field of the electromagnetic wave. However, external magnetic field increases the results in shrinking of the trajectories. Performed studies show that not only amplitude of the electric field but also the static axial magnetic field plays a crucial role in the acceleration process of a charged particle. At the authors of this paper best knowledge, the precise analytical solutions and theoretical analysis of the trajectories and energy gains by the charged particles accelerated in the laser beam and magnetic field are lacking in up to date publications. The authors have an intention to clarify partly some important aspects connected with this process. The presented theoretical studies apply for arbitrary charged particle and the attached figures-for electrons only.     

Variation in the magnetic structure of a single-domain particle ensemble and its response to an rf pulse

A set of equations in the Gilbert form that describes the motion of the magnetization vector in an ensemble of interacting magnetic nanoparticles is numerically solved for the case of high-amplitude rf pulses. Based on the numerical solution, the magnetic structure of spherical particle ensembles showing cubic anisotropy at certain parameters of the variable field is studied. This phenomenon is shown to have a threshold. The dependence of the field threshold amplitude on the acting pulse repetition rate and amount of magnetic interaction is determined. It is demonstrated that a change in the magnetic structure of the interacting particle ensembles causes a change in the spectrum of the response. This fact can be used for pulsed rf writing and readout of information based on ferromagnetic resonance.     

Operator method of integration of the equations of motion of a particle in an external field

A method is proposed for the determination of the laws of motion of a particle in an external field. Solutions are found for the equations of motion of a charged particle in the field of an undulator, and of the Dirac-Lorentz equation in a magnetic field. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 72–75, February, 1993.     

The classical and quantum mechanics of a particle on a knot

A free particle is constrained to move on a knot obtained by winding around a putative torus. The classical equations of motion for this system are solved in a closed form. The exact energy eigenspectrum, in the thin torus limit, is obtained by mapping the time-independent Schrödinger equation to the Mathieu equation. In the general case, the eigenvalue problem is described by the Hill equation. Finite-thickness corrections are incorporated perturbatively by truncating the Hill equation. Comparisons and contrasts between this problem and the well-studied problem of a particle on a circle (planar rigid rotor) are performed throughout.     

Dynamics of a charged particle in the presence of both a quasimonochromatic wave and a strong magnetic field

A multiple-frequency system of equations is averaged to obtain the relativistic equations of motion for a charged particle in the field of a quasimonochromatic wave propagating along a strong magnetic field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 70–72, December, 1976.We wish to thank L. M. Gorbunov for valuable discussions.