Propagator of a Charged Particle with a Spin in Uniform Magnetic and Perpendicular Electric Fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schrödinger equation and some special and limiting cases are outlined.     

Gyroscopically Stabilized Oscillators and Heat Baths

In this paper we analyze the stability of a gyroscopic oscillator interacting with a finite- and infinite-dimensional heat bath in both the classical and quantum cases. We consider a finite gyroscopic oscillator model of a particle on a rotating disc and a particle in a magnetic field and we examine stability before and after coupling to a heat bath. The heat bath is modelled in the finite-dimensional setting by a system of independent oscillators with mass. It is shown that if the oscillator is gyroscopically stable, coupling to a sufficiently massive heat bath induces instability even in the finite-dimensional setting. The key mechanism for instability in this paper is thus not induced by damping. The meaning of these ideas in the quantum context is discussed. The model extends the exact diagonalization analysis of an oscillator and field of Ford, Lewis, and O'Connell to the gyroscopic setting. We also discuss the interesting role that damping of Landau type plays in the infinite limit.     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics ofa particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

Non-equilibrium statistical mechanics of synchrotron-Cerenkov radiation

The synergic synchrotron-Cerenkov (SC) radiation emitted by a relativistic charged particle under the combined effect of the constant external magnetic field and the collective interactions in the ambient plasma (medium) is given in the framework of the non-equilibrium statistical mechanics developed by Prigogine and his co-workers. Starting from the formal solution of the Liouville equation, the one-particle distribution function is calculated. Restricting the motion of the test particle to a circular orbit in the plane normal to the magnetic field, we use the above distribution function to calculate the power emitted per unit solid angle by the test particle as a function of time. We have thus obtained the time evolution of the synergic SC radiation which in the asymptotic limit reproduces the results of Schwinger and his co-workers. It is also shown that the collective interactions within the system produces a shift in the frequency of the outcoming radiation.     

Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.     

Exact solution for a non-Markovian dissipative quantum dynamics

We provide the exact analytic solution of the stochastic Schr?dinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.     

Approach to statistical equilibrium in a soluble model

The soluble model of an oscillator coupled to a scalar field is used to describe the Brownian motion of an oscillator in a thermal bath. The approach to equilibrium is shown by studying the generating functional that corresponds to a non-equilibrium situation fixed as an initial condition. It is shown how this functional goes over into the functional associated with the equilibrium situation. The solution of the functional equations is discussed.     

Interaction between glow discharge plasma and dust particles

The effect of dust particle concentration on gas discharge plasma parameters was studied through development of a self-consistent kinetic model which is based on solving the Boltzmann equation for the electron distribution function. It was shown that an increase in the Havnes parameter causes an increase in the average electric field and ion density, as well as a decrease in the charge of dust particles and electron density in a dust particle cloud. Self-consistent simulations for a wide range of plasma and dust particle parameters produced several scaling laws: these are laws for dust particle potential and electric field as a function of dust particle concentration and radius, and the discharge current density. The simulation results demonstrate that the process of self-consistent accommodation of parameters of dust particles and plasma in condition of particle concentration growth causes a growth in the number of high-energy electrons in plasma, but not to depletion of electron distribution function.     

Quantum oscillator as 1D anyon

It is shown that, in one spatial dimension, the quantum oscillator is dual to the charged particle situated in the field described by the superposition of Coulomb and Calogero-Sutherland potentials.     

One Dimensional Models with a Singular Potential of the Type −<Emphasis Type="Italic">αδ</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)+<Emphasis Type="Italic">βδ</Emphasis>′(<Emphasis Type="Italic">x</Emphasis>)

We study a one-dimensional singular potential plus two types of regular interactions: constant electric field and harmonic oscillator. In order to search for the bound state energies, we shall use the Lippman-Schwinger Green function technique. Another direct method will be mentioned for the harmonic oscillator. In the electric field case the unique bound state coincides with that found in an earlier study as the field is switched off. For non-zero field the ground state is shifted and positive energy “quasibound states” appear. The harmonic oscillator demonstrates the general result that for a symmetric potential the odd states are not altered whereas the even states energies are lowered or raised accordingly as the delta perturbation is attractive or repulsive. No states are created or annihilated.     

Stochastic electrodynamics. III. Statistics of the perturbed harmonic oscillator-zero-point field system

In this third paper in a series on stochastic electrodynamics (SED), the nonrelativistic dipole approximation harmonic oscillator-zero-point field system is subjected to an arbitrary classical electromagnetic radiation field. The ensemble-averaged phase-space distribution and the two independent ensemble-averaged Liouville or Fokker-Planck equations that it satisfies are derived in closed form without furtner approximation. One of these Liouville equations is shown to be exactly equivalent to the usual Schrödinger equation supplemented by small radiative corrections and an explicit radiation reaction (RR) vector potential that is similar to the Crisp-Jaynes semiclassical theory (SCT) RR potential. The wave function in this SED Schrödinger equation is shown to have thea priori significance of position probability amplitude. The other Liouville equation has no counterpart in ordinary quantum mechanics, and is shown to restrict initial conditions such that (i) The Wigner-type phase-space distribution is always positive, (ii) in the absence of an applied field, the only allowed solution of both equations is the quantum ground state, and (iii) if a previously applied field is suddenly turned off, then spontaneous transitions occur, with no need for a triggering perturbation as in SCT, until the system is in the ground state. It is also shown that the oscillator energy is a fluctuating quantity that must take on a continuum of values, with average value equal to the quantum ground-state energy plus a contribution due to the applied classical field.     

Classical Model of Confinement

The confinement mechanism proposed earlier and then applied successfully to meson spectroscopy by one of the authors is interpreted in classical terms. For this aim the unique solution of the Maxwell equations, an analog of the corresponding unique solution of the SU(3)-Yang-Mills equations describing linear confinement in quantum chromodynamics, is used. Motion of a charged particle is studied in the field representing magnetic part of the mentioned solution and it is shown that one deals with the full classical confinement of the charged particle in such a field: under any initial conditions the particle motion is accomplished within a finite region of space so that the particle trajectory is near magnetic field lines while the latter are compact manifolds (circles). An asymptotical expansion for the trajectory form in the strong field limit is adduced. The possible application of the obtained results in thermonuclear plasma physics is also shortly outlined.     

Physical meaning of the stretched exponential Kohlrausch function

In this paper the physical meaning of the empirical Kohlrausch-Williams-Watts (KWW) function is explained in terms of the linear oscillator theory. It is shown that the KWW function is a solution of the non-autonomous linear first order equation for an overdamped linear oscillator. From the linear oscillator model it follows that the KWW-type relaxation is the linear relaxation with a time (coordinate, stress, voltage, etc.) dependent dissipation of energy. The theoretical results are validated by measurements. A method for modeling KWW-type relaxation using simple electrical circuits is proposed.     

Action principle of Bogoliubov coefficients

It is shown that the Schwinger action principle uniquely determines the Bogoliubov coefficients connecting in- and out-states for an oscillator with a time-dependent spring constant, and hence for a charged particle field in a constant homogeneous electric field. This allows a complete construction of the S-matrix in the external field approximation. Especially it allows an easy calculation of 〈out∥in〉, the vacuum-to-vacuum amplitude, in this approximation. The method is applied to electromagnetism embedded in a non-abelian gauge group.     

Justification of the “symmetric damping” model of the dynamical Casimir effect in a cavity with a semiconductor mirror*

A “microscopic” justification of the “symmetric damping” model of a quantum oscillator with time-dependent frequency and time-dependent damping is given. This model is used to predict the results of experiments on simulating the dynamical Casimir effect in a cavity with a photo-excited semiconductor mirror. It is shown that the most general bilinear time-dependent coupling of a selected oscillator (field mode) to a bath of harmonic oscillators results in two equal friction coefficients for the both quadratures, provided all the coupling coefficients are proportional to a single arbitrary function of time whose duration is much shorter than the periods of all oscillators. The choice of coupling in the rotating wave approximation form leads to the “minimum noise” model of the quantum damped oscillator, introduced earlier in a pure phenomenological way.     

The influence of a varying field on the non-Markovian migration of particles

The influence of an external varying field on the non-Markovian migration of particles described in the continuous-time random walk model (CTRWM) was analyzed theoretically. In terms of the Markovian representation for the CTRWM suggested earlier, a rigorous method for describing the influence of an external force was developed. This method reduced the problem to solving the non-Markovian stochastic Liouville equation (SLE) for the particle distribution function. An analysis of the derived SLE and its comparison with the earlier equations were performed. The method was used to study the characteristic features of the time dependence of the first and second moments of the distribution function for particles involved in subdiffusion motion in a uniform varying external field. Both oscillating and fluctuating fields were considered. In both cases, anomalously strong field effects on the second particle distribution moment (variance) were observed. This influence was especially strong for a fluctuating field, and in the limit of anomalously slow fluctuations at that.     

ANALYTICAL SOLUTION OF THE 2n-DIMENSIONAL FOKKER-PLANCK EQUATION WITH HARMONIC OSCILLATOR POTENTIAL

An analytical solution is obtained for the 2n-dimensiona Fokker-Planck equation (F-P equation for short) with the harmonic oscillator potential. A few steps are involved in the derivation. First,the Lagrangian subsidiary equation is solved; then with its integral constants as new variables of the F-P equation, the diffusion equation is obtained and solved; at last, expressed in the original phase space, the solution of the F-P equation .is finally obtained. The analysis for the solution is made. The solution is a Gaussian type function and a δ-function of time. If a particle moves in a well in ali directions, then as t→∞, the distribution function can reach a stationary nonzero distribution-Maxuwell-Boltzmann type distribution (M-B distribution for short).As an example, the 2-dimensional F-P equation is solved and discussed in detail.