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The angular distribution of polarization components of synchrotron radiation emitted by a charge moving along a spiral is investigated. The study is carried out within the framework of classical electrodynamics.

Synchrotron radiation was originally studied by classical methods using the Liénard–Wiechert potentials of electric currents. Subsequently, quantum corrections to the classical formulas were studied, considering the emission of photons arising from electronic transitions between spectral levels, described in terms of the Dirac equation. In this paper, an intermediate approach is considered, in which electric currents generating the radiation are considered classically while the quantum nature of the radiation is taken into account exactly. Such an approximate approach may be helpful in some cases; it allows one to study one-photon and multi-photon radiation without complicating calculations using corresponding solutions of the Dirac equation. Here, exact quantum states of an electromagnetic field interacting with classical currents are constructed and their properties studied. With their help, the probability of photon emission by classical currents is calculated and relatively simple formulas for one-photon and multi-photon radiation are obtained. Using the specific circular electric current, the corresponding synchrotron radiation is calculated. The relationship between the obtained results and those known before are discussed, for example with the Schott formula, with Schwinger calculations, with one-photon radiation of scalar particles due to transitions between Landau levels, and with some previous results of calculating two-photon synchrotron radiation.

We present a semiclassical approximation for treating the radiation from classical currents. In particular, we present exact quantum states of the quantized electromagnetic field interacting with classical currents. These states are used to calculate a probability of many-photon radiation from the vacuum initial state of the electromagnetic field. In this manner, in the present article, we study characteristics of electromagnetic radiation of a planar undulator. We find the total radiated energy and its spectral-angular distribution. We compare our results with ones obtained in the framework of classical electrodynamics, discussing differences introduced by accurate accounting for the quantum nature of electromagnetic radiation and present results of some numerical calculations that confirm, in particular, the latter discussion. In Appendix we present the calculation of the radiated energy using an alternative parametrization of the trajectory of electrons moving in a planar undulator.

An exact analytical expression for the effective angle is determined for an arbitrary energy value of a radiating particle. An effective angle of instantaneous power is defined for synchrotron radiation in the framework of classical electrodynamics. This definition explicitly contains the most symmetric distribution of half the total of the instantaneous power of synchrotron radiation. Two exact analytical expressions for the effective angle are considered for the arbitrary energy values of a radiating particle, and the second expression brings to light the exact asymptotics of the effective angle in the ultrarelativistic limit.

The instantaneous angular distribution of synchrotron radiation is investigated. The radiation space is divided into two parts. The first part of the space is the interior of a cone with the apex located at the point of a radiating charge, the angular span being $2\alpha_0$ and the central axis oriented along the instantaneous velocity of the charge. The second part of the space is the complement of the first part in the entire space. It is shown that the radiation in the ultrarelativistic limit is entirely contained in the first part of the space, and the radiation in the second part (with a non-zero $\alpha_0$) vanishes., Comment: 7 pages, 3 figures

The extension of the wave zone of synchrotron radiation is studied., 6 pages, 1 figure

A complete solution of the problem of squaring the Dirac equation with arbitrary external electromagnetic field is expounded.

The angular distribution functions of the polarized components of synchrotron radiation in the nonrelativistic approximation are investigated using methods of classical and quantum theory. Particles of zero spin (bosons) and spin 1/2 (electrons) are considered in the quantum theory. It is shown that in the first nonzero approximation the angular distribution functions, calculated by methods of classical and quantum theory, coincide identically. Quantum corrections to the angular distribution functions appear only in the subsequent approximation whereas the total radiated power contains quantum and spin corrections already in the first approximation.

We consider an example where coherent states appear naturally (dynamically). We study the interaction between an arbitrary quantized matter with a quantized electromagnetic field. Using an approximation that is similar to the Hartree–Fock one and a variational principle, we demonstrate that the electromagnetic field is created by the charged matter exactly in Glauber coherent states with parameters determined by mean currents of the matter.

We study properties of a particle moving in a double-well potential in the two-level approximation placed in an additional external time-dependent field. Using previously established property (J. Phys. A 41, 244023 (2008)) that any two-level system possesses a pseudo-supersymmetry we introduce the notion of pseudo-supersymmetric field. It is shown that these fields, even if their time dependence is not periodical, may produce the effect of localization of the particle in one of the wells of the double-well potential.

A physical interpretation of quantum numbers in the problem of charge motion in a constant uniform magnetic field is made.

A review of new achievements in theoretical investigations of angular distribution of synchrotron radiation power is presented.

For synchrotron radiation, the partial contribution is defined as the ratio of the power radiated by an individual spectral harmonic, ν = 1, 2, 3 …, to the total (summed over ν) radiated power. A complete analysis of the dependence of the partial contribution on ν, polarization of the radiation, and energy of the particle is given.

The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0+1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE., Comment: 29 pages

One of the remarkable features of the relativistic electron synchrotron radiation is its concentration in small angle Δ ≈ 1/ γ (here γ -relativistic factor: γ = E / mc 2 , E – energy, m – electron rest mass, c – light velocity) near rotation orbit plane [V.G. Bagrov, V.A. Bordovitsyn, V.G. Bulenok, V. Ya. Epp, Kinematical projection of pulsar synchrotron radiation profiles, in: Proceedings of IV ISTC Scientific Advisory Commitee Seminar on Basic Science in ISTC Aktivities, Akademgorodok, Novosibirsk, April 23–27, 2001, p. 293–300]. This theoretically predicted and experimentally confirmed feature is peculiar to total (spectrum summarized) radiating intensity. This angular distribution property has been supposed to be (at least qualitatively) conserved and for separate spectrum synchrotron radiation components. In the work of V.G. Bagrov, V.A. Bordovitsyn, V. Ch. Zhukovskii, Development of the theory of synchrotron radiation and related processes. Synchrotron source of JINR: the perspective of research, in: The Materials of the Second International Work Conference, Dubna, April 2–6, 2001, pp. 15–30 and in Angular dependence of synchrotron radiation intensity. http://lanl.arXiv.org/abs/physics/0209097 , it is shown that the angular distribution of separate synchrotron radiation spectrum components demonstrates directly inverse tendency – the angular distribution deconcentration relatively the orbit plane takes place with electron energy growth. The present work is devoted to detailed investigation of this situation. For exact quantitative estimation of angular concentration degree of synchrotron radiation the definition of radiation effective angle and deviation angle is proposed. For different polarization components of radiation the dependence of introduced characteristics was investigated as a functions of electron energy and number of spectrum component.

A complete investigation of the evolution of the angular distribution of circularly polarized synchrotron radiation with particle energy has been performed within the framework of the classical theory. To describe the familiar qualitative features of synchrotron radiation, exact quantitative characteristics have been introduced.

Darboux transformation method is applied to constructing new exactly solvable periodic pseudoscalar potentials for the one-dimensional Dirac equation. Factorization properties of Darboux transformation operators are established. Underlying quadratic supersymmetry is revealed.

A three-level system is considered as a simplest example where a possibility of solving the spectral problem is observed. We thoroughly analyze the relation between the amount of radiation emitted by the particle during the transitions to the first excited state and to the ground state. Generalizing basic expressions we can follow the evolution of spectral maximum. It turns out there is a condition for radiation maximum to stay at highest harmonic.

Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states., 10 pages, 1 figure. Published

Different families of generalized CS for one-dimensional systems with general time dependent quadratic Hamiltonian are constructed. In principle, all known CS of systems with quadratic Hamiltonian are members of these families. Some of the constructed generalized CS are close enough to the well-known due to Schr\"odinger and Glauber CS of a harmonic oscillator, we call them simply CS. However, even among these CS there exist different families of complete sets of CS. These families differ by values of standard deviations at the initial time instant. According to the values of these initial standard deviations one can identify some of the families with semiclassical CS. We discuss properties of the constructed CS, in particular, completeness relations, minimization of uncertainty relations and so on. As a unknown application of the general construction, we consider different CS of an oscillator with a time dependent frequency., Comment: 11 pages, 1 figures, Submitted to Brazilian Journal of Physics

Methods of the classical theory of relativistic radiation are used to investigate general properties of the total angular power distribution of radiation of a charge and to analyze the corresponding components of linearly polarized radiation emitted by this charge.

New experiments on channeling of 255 MeV electrons in a half-wavelength crystals (HWC) were performed at SAGA Light Source facilities. The simulations of trajectories for (2 2 0) and (1 1 1) planar channeling in Si were performed using the computer code BCM-1.0. Comparison of experimental and theoretical results shows a good agreement. The results of calculations of spectral distribution of radiation in forward direction ( θ = 0 ° ) from 255 MeV electrons at (2 2 0) channeling in HWC silicon are presented. Qualitative comparison with radiation spectrum from an electron moving in an arc is performed.

For an exact quantitative description of spectral properties of synchrotron radiation (SR), the concept of effective width of the spectrum is introduced. In the most interesting case, which corresponds to the ultrarelativistic limit of SR, the effective width of the spectrum is calculated for the polarization components, and new physically important quantitative information on the structure of spectral distributions is obtained. For the first time, the spectral distribution for the circular polarization component of the SR for the upper half-space is obtained within classical theory.

Aharonov-Bohm solenoid changes the energy spectrum of charge particles in pure magnetic field. In particular, the degeneracy with respect to azimuthal quantum number l is partially lifted. In turn, this complicates the radiation spectrum of a charged particle in magnetic field in the presence of the solenoid (Aharonov-Bohm effect). In particular, the degeneracy of the radiation intensity with respect to the azimuthal quantum number is lifted completely. In the present work we study l-dependence (induced by Aharonov-Bohm solenoid) of synchrotron radiation intensity in semiclassical approximation.

The second-order Darboux transform for a growing potential of a Schrodinger equation is discussed in detail. Restrictions on the transforming functions whose eigenvalues are greater than the energy of the ground state if the potential of the transformed Schrodinger equation is regular has been established. It has been shown that, along with the well-known elimination of two levels of the discrete spectrum, other opportunities can also be realized: (a) the transformed spectrum remains identical to the original one; (b) one level is eliminated from the spectrum; (c) one additional level is generated, and (d) two additional levels are generated.

We study the impact of Aharonov–Bohm solenoid on the radiation of a charged particle moving in a constant uniform magnetic field. Radiation peculiarities caused by the presence of the solenoid may be considered as a manifestation of Aharonov–Bohm effect in the radiation. In particular, new spectral lines appear in the radiation spectrum. Due to angular distribution peculiarities of the radiation intensity, these lines can in principle be isolated from basic cyclotron and synchrotron radiation spectra.

The dependence of spectral-angular distribution of synchrotron radiation (SR) from spin orientation has been investigated. The investigation presents that there are such features of the dependence that allow getting complete information about spin states of electron.

We consider the coherent states of a singular oscillator; these states are defined to be the characteristic states of an operator that reduces the number of basis functions for a discrete spectrum to one. We use the Darboux transform to study the coherent states of a transformed Hamiltonian. We obtain an expression for measures that can be used to decompose unity. We construct a holomorphic representation for the state vectors in the space of functions holomorphic everywhere in the complex plane, including vectors for discrete spectra and coherent states. We obtain a holomorphic representation of the Darboux transformation operators.

With the help of a nonstationary Darboux transformation, we have obtained a system of coherent states for the potential which is a solution of the Kadomtsev-Petviashvili equation. We have determined the measure realizing expansion of unity with respect to these states. We have constructed a holomorphic representation of the state vectors and the ladder operators for the discrete basis of the Hilbert space of solutions of the Schrodinger equation.

Darboux transformation is applied to three classical potentials, namely the harmonic oscillator, effective Coulomb and Morse potentials to generate exactly solvable potentials of elementary form. For every potential, the isospectral families of potentials are constructed. For almost all potentials, a set of normalized discrete spectrum wave functions is given.

We consider the problem of classification of Steckel spaces satisfying the system of self-consistent Einstein-Dirac equations with a cosmological term for the case when the spaces allow for isotropic complete sets of integrals of motion of type (3.1). The exact solution obtained contains four arbitrary functions of one variable.

The problem of exact integration of the Einstein–Dirac equation is studied. To solve this set of equations metrics admitting complete separation of variables in the Dirac equation are considered. At the first stage Stackel spaces of types (3.1) and (3.0) are studied. For the first case obtained solutions contain arbitrary functions depending on one variable only (five real functions for the Einstein–Weyl equations and four real functions for the Einstein–Dirac ones). For the second one all solutions belonging to the diagonal metrics of Bianchi type 1 are found.

The classification of isotropic conformal Steckel spaces satisfying a system of Einstein equations in which the right-hand side is the energy—momentum tensor of an isotropic ideal liquid is considered. The complete solution of the problem is found for the case of a conformal Steckel space admitting of one isotropic Killing vector field and two Killing tensor fields, when these objects form a complete set.

Two kinds of transformation for the time-dependent Schrodinger equation, i.e. the differential and integral transformations, are introduced. If one considers only stationary solutions of this equation, both transformations reduce to the well known Darboux transformation for the stationary Schrodinger equation. When applied to non-stationary solutions, they give different results. Both transformations are invertible in appropriate spaces. With the help of these transformations alternative systems of coherent states to those in the literature are obtained for isospectral Hamiltonians with equidistant spectra. These transformations are also applied to the construction of coherent states for Hamiltonians whose spectrum consists of an equidistant part and one separately disposed level with an energy gap equal to the k skipped levels.