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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.

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.

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.

A concept of semiclassically concentrated states is developed on the basis of the Maslov germ theory. Higher approximations of semiclassical trajectory-coherent states and of semiclassical Green function (in the class of semiclassically concentrated states) for a many-dimensional Schrodinger-type equation are constructed. It is shown that, in class of such semiclassically concentrated states, a Schrodinger-type equation (up to any order of ℏ, ℏ→0) is equivalent, from the viewpoint of calculating the quantum averages, to a closed finite system of ordinary differential equations.

The supersymmetry of a one-dimensional time-dependent Schrodinger equation is established. It is intimately connected with the time-dependent Darboux transformation. With the help of this transformation new exactly solvable time-dependent potentials are generated.

Recently, an implementation of a universal set of one- and two-quantum-bit gates for quantum computation using spin states of coupled single-electron quantum dots was proposed. It was demonstrated that it is possible to execute a coherent control of a quantum system based on two-electron spin states in a double quantum dot, allowing state preparation, coherent manipulation, and projective readout. This possibility is based on rapid electrical control of the spin exchange interaction. These results motivated us to develop a formal theoretical study of the corresponding model of two coupled spins placed in a magnetic field and subjected to a time-dependent mutual Heisenberg interaction. Using possible exact solutions of the corresponding quantum problem, we study entangling of different separable initial states in this model. It is demonstrated that the entanglement due to a time-dependent Heisenberg interaction is dominating in comparison with the entanglement due to the action of an external magnetic field.

We consider an example of a quantum system with a continuous and unbounded spectrum—a particle in the parabolic potential barrier. Conditionally, we call this system the inverse oscillator. Following the method used in our previous publication (Bagrov et al 2012 J. Phys. A: Math. Theor. 45 125306), we construct different families of generalized coherent states (GCS). We discuss properties of the constructed GCS and their relation to what is already known in the literature about semiclassical states for the inverse oscillator. Then in the same manner, we construct families of GCS for the normal oscillator. Using special variables, we succeeded in constructing GCS for the normal oscillator that admit different limiting cases: free particle GCS, the usual Schrodinger, Glauber CS and GCS of the inverse oscillator.

Examining the characteristics of radiation from scalar particles in second excited state, which could be regarded at as a three-level system, we deliver (and solve) the problem on spectral properties in simplest form. Using the methods of quantum theory, we establish that partial contribution of fixed harmonic to the total power of radiation does not vanish in ultrarelativistic domain, and this fact contradicts the results known in classical approach. We also consider the evolution of radiation maximum. In particular, for a scalar particle we find the condition for radiation maximum to stay at highest possible harmonic.

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.

On the basis of Maslov's complex germ method for the Dirac operator in external electromagnetic and torsion fields the quasi-classical spectral series corresponding within the limit h(cross) to 0 to the electron motion along closed stable orbits has been constructed. The quasi-classical energy spectrum is found from the condition of quantization of these orbits, and the quasi-classical asymptotics corresponding to the latter form a complete set of localized quantum state. The method is illustrated in all details by the electron motion in the axial fields as an example.

It is shown that the semiclassical approximation in quantum mechanics is equivalent to replacement of the Schrodinger equation by a finite closed system of first-order ordinary differential equations with initial conditions that satisfy special restrictions.

An integration method for the Dirac equation is proposed. The method, based on diagonalization, reduces the problem to one of integration of independent second-order differential equations.

The first-order quantum correction for the characterization of spontaneous radiation is calculated by means of electron quasi-classical trajectory-coherent states in an arbitrary electromagnetic field. Well known expressions for the characterization of spontaneous radiation are obtained using quasi-classical approximation. The first-order quantum correction is derived as a functional from a classical trajectory (among which is a classical spin vector). Transitions with spin flip and without spin flip are distinguished. Those elements connected with photon kick and quantum motion characteristics are selected for first-order quantum correction. It is shown that, using an ultra-relativistic approximation, the latter may be ignored, but when using a non-relativistic approximation their contributions are approximately equal. A special trajectory-coherent representation that significantly simplifies the investigation of spontaneous radiation is proposed.

The concept of quasiclassically localized states is defined and the theorem about the possibility of the quasiclassical localization on the phase trajectories only (solutions of corresponding classical equations of Hamilton) is proved.

The authors present the trajectory-coherent approximation for the Proca equation in the Riemann-Cartan space with an external electromagnetic field. The scheme for constructing the special class of asymptotical quasiclassical trajectory-coherent states (TCS) is given. They are localized in the neighbourhood of the wordline of the charged particle and admit the Hilbert structure with respect to the Proca inner product. The problem of the TCS construction is reduced to solving the Pauli-type linear evolution equation for the polarization vector. The pseudovector of the spin as the quantum mechanical average of the suitable spin operator in the TCS is defined and the covariant generalization for the well known Bargmann-Michel-Telegchi spinning equation in the case of background torsion fields from the Proca equation is obtained.

A semiclassically concentrated state in quantum mechanics is defined. For the example of a one-dimensional Schrodinger equation, a theorem is proved which shows that the condition of semiclassical concentration can be realized only on the solutions of the corresponding classical system of Hamilton equations.

For pt.I see ibid., vol.7, p.517, (1990). This is the second paper in a series of research started by Bagrov et al. The Abelian complete sets of matrix first-order differential symmetry operators for the Dirac equation are used for the separation of variables in the Dirac equation in Stackel spaces with external gauge fields. The permissible space metrics and gauge field potentials are not limited by field equations.

The problem of complex separation of variables in the wave equation is considered in four-dimensional Minkowskii space-time. In contrast to the known series of researches by Kalnins and Miller (see Ref. Zh., Fiz., 2B9 (1978); 1B208 and 1B209 (1979), e.g.), underlying this research is a theorem on the necessary and sufficient conditions of total separation of variables in the non-parabolic V. N. Shapovalov equation (Differents. Uravn.,16, No. 10, 1864–1874 (1980)). Nonequivalent complete sets of three differential first-order symmetry operators are constructed, appropriate coordinate systems are found, and complete separation of variables is performed in the wave equation.

The subspaces of Riemannian space of signature (+---) that admit separation of the Dirac equation have been found in the case of Riemannian space admitting the separation of the Hamilton-Jacobi equation. For the separation of variables in the Hamilton-Jacobi equation it is necessary for the complete set of Killing vectors and tensors to be of a special kind. Every complete set defines its own type of metric of Riemannian space which is called Stackel space. The Dirac equation does not permit the separation of variables in general cases of Stackel space. The main idea of the paper is in the construction, in Stackel space, of a complete set of another kind. This complete set consists of three matrix first-order differential symmetry operators of the Dirac equation. The operators are pairwise commuting and linearly independent. The complete set structure is in agreement with the structure of the Killing vectors and tensors of Stackel space. The separation of variables in the Dirac equation has been carried out in the explicit form in Stackel spaces which admit complete sets of symmetry operators. These operators have been used essentially in the process of separation that differs from Chandrasekhar method.

V.G. Bagrov a, A.A. Pecheritsyn b , B.F. SamsonovC Physics Faculty of Tomsk State University, 634050 Tomsk, Russia Abstract. The Darboux transformation operator method is extended to the onedimensional stationary Dirac equation. This permits one, starting from an exactly solvable Dirac equation, to obtain a family of matrix-valued exactly solvable Dirac potentials. New potentials can be isospectral with the initial one or their spectra differ by one or two levels. Special cases of scalar and pseudoscalar potentials for which the Dirac equation is equivalent to a supersymmetric pair of Schrodinger equations are analyzed in details. An interrelation between Schrodinger and Dirac transformation operators is found. It is shown that the Dirac case induces the transformations of either supersymmetric Schrodinger partners.

The radiation of distributed (non-pointlike) charges is studied. The role of internal structure is analyzed. It is shown that, in general, the structure effects give rise to damping of the high-frequency part of the spectrum. Some interesting examples are considered in detail. It is found in particular that rotating systems can generate monochromatic radiation.

A new approach to constructing coherent states (CS) and semiclassical states (SS) in magnetic-solenoid field is proposed. The main idea is based on the fact that the AB solenoid breaks the translational symmetry in the xy-plane, this has a topological effect such that there appear two types of trajectories which embrace and do not embrace the solenoid. Due to this fact, one has to construct two different kinds of CS/SS, which correspond to such trajectories in the semiclassical limit. Following this idea, we construct CS in two steps, first the instantaneous CS (ICS) and the time dependent CS/SS as an evolution of the ICS. The construction is realized for nonrelativistic and relativistic spinning particles both in (2+1)- and (3+1)- dimensions and gives a non-trivial example of SS/CS for systems with a nonquadratic Hamiltonian. It is stressed that CS depending on their parameters (quantum numbers) describe both pure quantum and semiclassical states. An analysis is represented that classifies parameters of the CS in such respect. Such a classification is used for the semiclassical decompositions of various physical quantities., 35 pages, 2 figures. Some typos in (77), (101), and (135) corrected with respect to the published version. Results unchanged

We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein-Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov-Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special non-uniform fields. The solutions obtained can be useful to study Aharonov-Bohm effect in the corresponding electromagnetic fields., Comment: 39 pages, LaTex file

The search for new exact solutions to the Dirac and Klein-Gordon equations initiated in [1] is continued. New solutions are found for axisymmetric fields and one type of nonstationary field of special configuration. The basic notation and system of units of [1] are retained.

The Dirac equation is solved exactly for an electron interacting with both the quantum field of a plane wave and with a classical electromagnetic field, consisting of a plane wave and a constant, homogeneous magnetic field, parallel to the direction of propagation of both waves.

The one-loop effective action for some variants of d -dimensional quantum R 2 -gravity and for the special model of multidimensional gravity on the background M 4 × T d −4 , where M 4 is the Minkowski space, T d −4 is the ( d - 4)- dimensional torus, is obtained. It is shown that the effective action of the R 2 -gravity without the cosmological and Einstein terms leads to instability of classical compactification. It is shown by numerical calculation that the effective action of five-dimensional R 2 -gravity with the cosmological term admits self-consistent spontaneous compactification. The one-loop effective action for the five-dimensional Einstein gravity with antisymmetric torsion on the background M 4 × S 1 is also found.

A connection is discussed between the group SU(2) and supersymmetry for a series of quantum mechanical problems. It is pointed out that the impossibility of factorizing Hamiltonians obtained based on representations of the group SU(2) indicates that the supersymmetry of the system is broken.

A system of Einstein-Maxwell equations is integrated for the metric that generalizes the metric of a space with full set of type (1.1). This is necessary for carrying out the classification of the corresponding Stackel electrovacuum spaces. An algebraic classification of the solutions obtained is carried out.

The first part of the present investigation attempts to clarify some unsolved theoretical problems for the well-known model — electron plus a secondarily quantized plane wave. Approximations are found to bring this model within the framework of exact quantum electrodynamics. Some mathematical and physical features of the problem are studied.

Nonequivalent complete sets of first-order symmetry operators of the Dirac free equation determine the Yang-Mills field, permitting complete variable separation in the Dirac equations with an external Yang-Mills field. Typical representatives of the classes of permissible fields are considered.

The Dirac equation is solved for an electron moving in a quantized plane-wave field in the classical field of a longitudinal traveling electric wave propagating in one direction. Through a canonical transformation of the photon creation and annihilation operators the problem is reduced to a quasiparticle problem; the quasiparticle energy depends on the time and the coordinates.

Some general properties of coherent states of a particle in quantum mechanics are discussed and examples of relativistics coherent states of an electron moving in a steady and homogeneous field and in a non-stationary external field of special kind are presented. In particular cases this field may be a plane wave, a Redmond field (a superposition of plane wave and homogenous magnetic field), a superposition of a Redmond field and a longitudinal electric field or other physically interesting fields. Uber koharente Zustande relativistischer Partikel Im folgenden werden einige allgemeine Eigenschaften koharenter Zustande eines Partikels in der Quantenmechanik betrachtet und Beispiele relativistisch koharenter Zustande eines Elektrons in einem stationaren homogenen Feld sowie in einem nicht-stationaren Feld eines speziellen Typs angegeben. Insbesondere kann dieses Feld ein Redmond-Feld (eine Superposition einer ebenen Welle und eines homogenen Magnetfeldes), eine Superposition eines Redmond-Feldes und eines longitudinalen elektrischen Feldes oder ein anderes Feld von physikalischem Interesse sein.

The Dirac equation is solved exactly for an electron interacting with a quantized and a classical plane wave traveling in the same direction. It is shown that in a certain sense the system can be regarded as consisting of noninteracting quasiparticles.

Classification of all electrovac specetimes permitting the separation of variables in the Hamilton-Jacobi equation for a charged test particle is carried out. This separation requires the existence of a complete set consisting of Killing’s vectors and tensors of a special kind. Every complete set defines its own type of metric and electromagnetic potential in the separable coordinate system. There exist seven types of separation of variables for electromagnetic spaces. For every type an additional classification is carried out by transformation of coordinates without any disturbance of the separation conditions, the gradient transformation of electromagnetic potential and the conformal-constant transformation of metric. The key step in solving the problem is the extraction of an autonomous subsystem which determines the metric from only the Einstein-Maxwell equations for every type of separation of variables. Representatives of all classes of metrics and electromagnetic potential are given for every type of separation of variables with the exception of the spaces found in the well-known work by Carter. The problem is solved in terms of metric formalism. The classes of electrovac spacetimes obtained are found to be related to Petrov’s classification.

The problem of the construction of coherent states of relativistic particles (bosons and fermions) in electromagnetic fields has been solved. The states discovered are exact solutions to the Klein-Gordon and Dirac equations; the wavefunctions of the particles are found in the homogeneous magnetic and electric fields and the Redmond configuration field, as well as in the plane-wave field. The formulation of relativistic quantum mechanics is used on the 'null' plane.

An attempt is made to construct an equation analogous to the Dirac equation, but which has chiral symmetry. The most natural equation of this type is a nonlinear equation.

A general theory is derived for the creation of pairs of spinless particles from vacuum in the presence of an electromagnetic field. If the exact solutions of the Klein-Gordon equation are known for the particle in the given external field, the pair-creation problem can be solved.