Quantum Solutions in Relativistic Classical Mechanics

Russian Physics Journal - Quantum solutions of the classical equation of relativistic mechanics have been found. A synthesis of classical and quantum physics can become a basic formalism for a...     

Selected Problems of Relativistic Quantum Mechanics and Atomic Physics

The paper presents a brief review of the scientific work performed by the authors in the field of quantum mechanics and atomic, laser, and mathematical physics. The following problems are considered: the semiclassical theory of tunneling and multiphoton ionization of atoms and ions in a strong electromagnetic field; generalization of the Keldysh ionization theory to the relativistic case; calculation of the Coulomb corrections to the ionization rate of atoms for arbitrary values of the adiabaticity parameter γ: from γ ≪ 1 (the adiabatic region) to γ ≫ 1, when the laser field changes its direction and magnitude many times during the time of flight of the electron through the barrier; the Lorentz ionization of atoms moving in a constant magnetic field; the WKB approximation and the imaginary time method for describing electron tunneling through a time-varying barrier; the Stark effect in a strong field; the energy spectrum of a hydrogen atom in a strong and superstrong magnetic field; quantization with account of the barrier transparency; creation of electron-positron pairs from vacuum in a constant electric or intense pulsed (laser) field and the dependence of the number of pairs on the intensity and frequency of the laser field; the Feynman method of disentanglement of noncommuting operators and its applications: transitions between atomic states in an alternating magnetic field (the Majorana problem); a quantum oscillator with time-dependent frequency; and a singular oscillator. The mathematical problems of quantum mechanics are considered: the fall of a particle to the center; modification of the Bohr-Sommerfeld quantization condition for potentials with a barrier and the Kramers matching conditions; divergence of perturbation series and their summation; eigenvalues of the Casimir operators for irreducible representations of Lie groups, including the SU(2), SU(3), and SU(6) groups, which are widely used in physics.     

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.     

The Quasi-Exactly Solvable Problems in Relativistic Quantum Mechanics

We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein-Gordon and Dirac equations with equal vector and scalar potentials, and try to find the general form of the quasi-exactly solvable potential. After obtaining the general form of the potential, we present several examples to give the specific forms. In the examples, we show for special parameters the harmonic potential plus Coulomb potential, Killingbeck potential and a quartic potential plus Cornell potential are quasi-exactly solvable potentials.     

Geometrizing Relativistic Quantum Mechanics

We propose a new approach to describe quantum mechanics as a manifestation of non-Euclidean geometry. In particular, we construct a new geometrical space that we shall call Qwist. A Qwist space has a extra scalar degree of freedom that ultimately will be identified with quantum effects. The geometrical properties of Qwist allow us to formulate a geometrical version of the uncertainty principle. This relativistic uncertainty relation unifies the position-momentum and time-energy uncertainty principles in a unique relation that recover both of them in the non-relativistic limit.     

Some New Exact Solutions of mKdV Equations

New exact solutions are determined to the coupled mKdV equations by means of modified mapping method.     

Solutions of Fractional Partial Differential Equations of Quantum Mechanics

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox's $H$-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177-190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.     

Relativistic Quantum Mechanics and Field Theory

The problems which arise for a relativistic quantum mechanics are reviewed and critically examined in connection with the foundations of quantum field theory. The conflict between the quantum mechanical Hilbert space structure, the locality property and the gauge invariance encoded in the Gauss' law is discussed in connection with the various quantization choices for gauge fields.     

Relativistic Quantum Mechanics as a Telegraph

A derivation by Fröhner of non-relativistic quantum mechanics via Fourier analysis applied to probability theory is not extendable to relativistic quantum mechanics because Schrödinger's positive definite probability density * is lost (Dirac's spin 1/2 case being the exception). The nature of the Fourier link then changes; it points to a redefinition of the probability scheme as an information carrying telegraph, the code of which is Born's as extended by Dirac and by Feynman. Hermitian symmetry of the transition amplitude between Dirac representations expresses reciprocity of preparation and measurement (the quantal coding and decoding), two equally active interventions of the physicist; as the measurement perturbs the system retrodiction implies retroaction evidenced in delayed choice. Reciprocity of knowledge and organization vindicates Wigner's claim that reciprocal to the action of matter upon mind there exists a direct action of mind upon matter: psychokinesis, branded by Jaynes as a psychiatric disorder of the Copenhagen school. As for factlike irreversibility, it is expressed by the enormity of the change rate from information to negentropy: while gain in knowledge is normal psychokinesis is paranormal. Stapp's recent discussion of psychokinesis in a quantum context should be resumed in association with an EPR correlation; an experimental test is proposed.     

From Electromagnetism to Relativistic Quantum Mechanics

We study the relationship between Maxwell and Dirac equations for a class of solutions of Maxwell equations that can represent purely electromagnetic particles.     

Time Operator in Relativistic Quantum Mechanics

It is first shown that the Dirac's equation in a relativistic frame could be modified to allow discrete time, in agreement to a recently published upper bound. Next, an exact self-adjoint 4 × 4 relativistic time operator for spin-1/2 particles is found and the time eigenstates for the non-relativistic case are obtained and discussed. Results confirm the quantum mechanical speculation that particles can indeed occupy negative energy levels with vanishingly small but nonzero probablity, contrary to the general expectation from classical physics. Hence, Wolfgang Pauli's objection regarding the existence of a self-adjoint time operator is fully resolved. It is shown that using the time operator, a bosonic field referred here to as energons may be created, whose number state representations in non-relativistic momentum space can be explicitly found.     

Relativistic Quantum Mechanics and the Bohmian Interpretation

No Heading Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions.     

Lienard Equation and Exact Solutions for Some Soliton-Producing Nonlinear Equations

In this paper, we first considerexact solutions for Lienard equationwith nonlinear terms of any order.Then, explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq, generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.     

Cluster Separability in Relativistic Quantum Mechanics

Using a simple model we provide a quantitative study of the size of the corrections needed to restore cluster properties to the construction of Poincaré invariant dynamical models with kinematic spins, first provided by B. Bakamjian and L. H. Thomas. Our model calculations suggest that these corrections are too small to have a quantitative impact on nuclear physics observables calculated using models with meson and nucleon degrees of freedom.     

Systems of Covariance in Relativistic Quantum Mechanics

Developing some earlier work for spin-zerosystems found in the literature, we use some recentlyobtained generalized systems of covariance for thePoincare group to suggest a method for definingcovariant localization operators on phase space formassive relativistic particles with arbitrary integralor half-integral spins. These operators lead tooperationally defined position operators on spacelikehyperplanes, which turn out to be the Newton-Wigneroperators, and, as in the earlier results on spin-zerosystems, admit a consistent probability interpretationwith conserved currents.     

Formal Linearization and Exact Solutions of Some Nonlinear Partial Differential Equations

Abstract

An efficient method for constructing of particular solutions of some nonlinear partial differential equations is introduced. The method can be applied to nonintegrable equations as well as to integrable ones. Examples include multisoliton and periodic solutions of the famous integrable evolution equation (KdV) and the new solutions, describing interaction of solitary waves of nonintegrable equation.     

Causality and Transition Amplitude in Relativistic Quantum Mechanics

The constraint from causality on transition amplitude in relativistic quantum mechanics (x, γ|0, o)=0 when γ²-|x|²<0 is proven for Dirac particle without and with interactions with external fields and for free scalar and vector particle in any (1+D)-dimensional Minkowskian space-time.     

Introducing Relativistic Quantum Mechanics to Energy Students

A method for introducing relativistic quantum mechanics to energy students is described. The method complements existing modern physics courses and relies on Feynman’s relativistic path integral approach to display a relationship between classical dynamics, quantum theory, and relativistic quantum theory.