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Mathematical Problems in Engineering

Dynamical Aspects of Macroscopic and Quantum Transitions due to Coherence Function and Time Series Events

Ezzat G. Bakhoum, Cristian Toma

Published Jan 1, 2010• 35 citations
Publisher

Abstract

This study presents the application of dynamical equations able to generate alternating deformations with increasing amplitude and delayed pulses in a certain material medium. It is considered that an external force acts at certain time interval (similar to a time series) upon the material medium in the same area. Using a specific differential equation (considering nonzero initial values and using a function similar to the coherence function between the external force and the deformations inside the material), it results that modulated amplitude oscillations appear inside the material. If the order of the differential dynamical equation is higher, supplementary aspects as different delayed pulses and multiscale behaviour can be noticed. These features are similar to non‐Markov aspects of quantum transitions, and for this reason the mathematical model is suitable for describing both quantum phenomena and macroscopic aspects generated by sequence of pulses. An example of a quantum system, namely, the Hydrogen atom, is discussed.

DOI
10.1155/2010/428903

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This study presents the application of dynamical equations able to generate alternating deformations with increasing amplitude and delayed pulses in a certain material medium. It is considered that an external force acts at certain time interval (similar to a time series) upon the material medium in the same area. Using a specific differential equation (considering nonzero initial values and using a function similar to the coherence function between the external force and the deformations inside the material), it results that modulated amplitude oscillations appear inside the material. If the order of the differential dynamical equation is higher, supplementary aspects as different delayed pulses and multiscale behaviour can be noticed. These features are similar to non‐Markov aspects of quantum transitions, and for this reason the mathematical model is suitable for describing both quantum phenomena and macroscopic aspects generated by sequence of pulses. An example of a quantum system, namely, the Hydrogen atom, is discussed.
This study presents an alternating coordinate-momentum representation for propagation and transition of associated wave function, based on Bopp operators and on a certain symbolic determinant corresponding to a set of two linear equations with null free terms. It is shown that this alternating representation can justify in a good manner the patterns created through reflection/refraction of waves on nonperfectly smooth interfaces and phase correspondence of diffracted beams without the need of supplementary support functions. Correlations with Lorentz transformation of wave functions by interaction with a certain material medium (the space-time origin of a wave-train being adjusted) are also presented, and supplementary aspects regarding the use of electromagnetic scalar and vector potentials for modelling evolution within this alternating representation are added.
By analogy with the mathematical definition of degenerate systems, it is suggested that systems which are interrupted before they have traversed a full period, or which are in an external field which varies greatly in the time of one period, be called physically degenerate. It is further suggested, in harmony with the ideas of Ehrenfest and Tolman, that such systems show weak quantization, the quantization becoming less and less complete as the degeneracy increases. In the case of partially degenerate systems, the values of the related action variable and of the energy are grouped more or less closely about the mean values which they would all have if the systems were completely quantized. As examples, these ideas are applied to several problems, such as quantization in a small or rapidly changing magnetic field (including Glaser's results for the change of diamagnetic susceptibility of some gases with pressure, experiments by Breit and Ellett and by Wood and Ellett on depolarization, and the experiment of Gerlach and Stern in which atoms are projected into a strong field), absorption in the higher lines of the principal series of an alkali (including a discussion of quantization in hyperbolic and parabolic obits), and the continuous x-ray spectrum. It is then pointed out that this assumption demands the existence of quantum forces which have not been previously considered in quantum theory and whose action is to stabilize the stationary states, bringing the action variables and energy rapidly nearer and nearer to the proper values as time goes on. The suggestion is made that these forces are responsible for the large change of energy and action variables during a quantum transition. From this point of view the adiabatic theorem appears in a simple light; when the external parameters vary too rapidly, the quantization becomes poor, the quantum forces become strong and put the atom into a quantized state again. In any atom but hydrogen, quantum forces must be continuously acting to oppose the interchange of energy and momentum between electrons and to keep each electron quantized. These forces thus may prove to be important in the solution of problems of the quantum dynamics of the constitution of atoms, though they must be considered in connection with the dynamics of the oscillators.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
Recently, the possibility of the generation and interaction of unipolar half-cycle electromagnetic pulses with quantum systems has been the subject of active research. Such pulses can have many different and interesting applications. They are able to excite quantum systems very fast. Based on the numerical solution of Maxwell-Bloch equations, this paper theoretically studies the possibility of guiding and ultrafast controlling population difference gratings by a sequence of half-cycle attosecond pulses in a three-level resonant medium. The parameters of the model medium (transition frequencies and dipole moments of the transitions) are close to those found in the hydrogen atom. We show the possibility of guiding periodic gratings and dynamic microcavities on different resonance transitions of the medium. In this case, the medium considered is an example of a spatiotemporal photonic crystal. The refractive index varies rapidly in space and time. The possibility of the use of such gratings as Bragg mirrors in ultrafast optics is discussed. The results obtained are of interest in two actively developing areas of modern optics - the physics of half-cycle pulses and space-time photonic crystals.
Some phenomena in physics (photonic echo) appears for an external observer as non-causal pulses suddenly emerging from an active medium (prepared by some other optical pulses) being hard to be simulated. It is shown that such functions can be put in correspondence with acausal pulses in physics (generated by the wave-equation considered on the length interval (0, 1) (an open set), starting at the initial moment of time from null initial conditions). Consequences for a possible control of such phenomena are also presented.
We investigate quantum-mechanical counterpart of a classical instability in a phase space by the numerical method of quantum trajectories with moving basis. As an application the model of coupled two oscillators driven by a monochromatic force in the presence of dissipation (intracavity second harmonic generation) is analyzed. The system of interest is characterized by two bifurcations leading to ranges of instability: the Hopf bifurcation which connects a steady state dynamics of the oscillatory modes to a self-pulsing temporal dynamics and the bifurcation of the period-doubling. The both two regimes are analyzed on the framework of the semiclassical phase trajectories and the Wigner functions of the oscillatory modes in phase space.