Theoretical formalism

Kaplunovsky and Weinstein [kaplu85j develop a field-theoretic formalism that treats the topology and dimension of the spacetime continuum as dynamically generated variables. Dimensionality is introduced out of the characteristic behavior of the energy spectrum of a system of a large number of coupled oscillators. 

ANALYSIS OF NONLINEAR OPTICAL ACTIVITY 3.1 Theoretical Formalism... 

To simplify the theoretical formalism, we assume that the thickness of the... 

A theoretical formalism is available for understanding optical charge transfer processes in a variety of chemical systems (mixed-valence ions, donor-acceptor complexes, metal-ligand charge transfer chromophores, etc) where the extent of charge transfer is large and where electronic coupling between the electron donor and acceptor sites is relatively small. 

Various theoretical formalisms have been used to describe chemical exchange lineshapes. The earliest descriptions involved an extension of the Bloch equations to include the effects of exchange [1, 2, 12]. The Bloch equations formalism can be modified to include multi-site cases, and the effects of first-order scalar coupling [3, 13, 24]. As chemical exchange is merely a special case of general relaxation theories, it may be compre-... 

Chapter F presents the most advanced theory of 5 f bonding. An effort was made by the author (as in the rest of the book) to present the essential theoretical formalism in a simplified and hopefully understandable manner. At least, the reader should be able to find in this chapter a help to introduce himself to the most refined theoretical treatments which are nowadays of great relevance in actinide solid state physics. 

In order to describe strong-field interaction of the five-state system in Figure 6.9 with intense shaped femtosecond laser pulses, the theoretical formalism prepared in Section 6.3.2.1 is readily extended. The RWA Hamiltonian H f) for the five-state system in Figure 6.9 in the frame rotating with the carrier frequency reads... 

The use of the same analogy for the A + B - C reaction, described by a set of (2.3.67) is more problematic coupling of these equations results in a non-conserving number of particles in a system. This problem could be much easier treated in terms of the field-theoretical formalism. 

The use of an elegant field-theoretical formalism has been already discussed in Section 2.3.2 (see also review articles and a book [11, 41, 42] and... 

Note that another consisitent approach to the problem of mobile particle accumulation is based on the field-theoretical formalism [15, 37, 51]. However by two reasons this approach is not useful for the study of immobile particle aggregation (i) the smallness of the parameter U(t) = n(t)vq [Pg.414]

Of special interest in the recent years was the kinetics of defect radiation-induced aggregation in a form of colloids-, in alkali halides MeX irradiated at high temperatures and high doses bubbles filled with X2 gas and metal particles with several nanometers in size were observed [58] more than once. Several theoretical formalisms were developed for describing this phenomenon, which could be classified as three general categories (i) macroscopic theory [59-62], which is based on the rate equations for macroscopic defect concentrations (ii) mesoscopic theory [63-65] operating with space-dependent local concentrations of point defects, and lastly (iii) discussed in Section 7.1 microscopic theory based on the hierarchy of equations for many-particle densities (in principle, it is infinite and contains complete information about all kinds of spatial correlation within different clusters of defects). 

Equations (7.3.23) and (7.3.24) actually imply that one- and two-dimensional cases actually exhibit already macroscopic separation of the system into regions consisting of only A particles and only B particles. This is also confirmed by the fact that the integral over the spectrum of spatial fluctuations diverges in the cases at small k. On the other hand, to find the aggregation of particles in numerical experiments in the f/iree-dimensional case we must treat the deviations from the Poisson distribution in large volumes. More detailed field-theoretical formalism has confirmed this conclusion [15]. 

In spite of simple theoretical formalism (for example, mean-field descriptions of certain aspects) structural aspects of the systems are still explicitly taken into account. This leads to the results which are in a good agreement with computer simulations. But the stochastic model avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed for obtaining good statistics for the reliable results. Therefore more complex systems can be studied in detail which may eventually lead to better understanding of real systems. In the theory discussed below we deal with a disordered surface. This additional complication will be handled in terms of the stochastic approach. This is also a very important case in catalytic reactions. 

The analysis of the transient fluorescence spectra of polar molecules in polar solvents that was outlined in Section I.A assumes that the specific probe molecule has certain ideal properties. The probe should not be strongly polarizable. Probe/solvent interactions involving specific effects, such as hydrogen-bonding should be avoided because specific solute/solvent effects may lead to photophysically discrete probe/solvent complexes. Discrete probe/solvent interactions are inconsistent with the continuum picture inherent in the theoretical formalism. Probes should not possess low lying, upper excited states which could interact with the first-excited state during the solvation processes. In addition, the probe should not possess more than one thermally accessible isomer of the excited state. 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

We described so far the mathematical apparutus which will be used to obtain a group-theoretical formulation of quantum mechanics (section 3), by means of the Heisenberg group, and to obtain the connection between quantum and classical mechanics (section 4) within the group-theoretical formal-... 

The group-theoretical formalism we have introduced so far is particularly suited to formulate quantum mechanics in the Heisenberg representation, where the time dependence is shifted from the wavefunctions to the operators. As we shall show in the next section, the formalism allows to show in a straightforward way that the Poisson brackets are obtained as a formal limit of the commutator when h —> 0. 

In the present and in the following section we discuss the application of the group-theoretical formalism to the formulation of quantum-classical mechanics. Our purpose is to determine evolution equations for two coupled subsystems, with two different degrees of quantization. We have shown in the previous sections that the classical behaviour of a system is formally obtained as a limiting case of the quantum behaviour, when the Planck constant h tends to zero. In this section we will associate two different values of the Planck constant, say hi and /12, to the two subsystems and introduce suitable Lie brackets to determine the evolution of the two subsystems [15]. The consistency, e.g., with respect to Jacobi identity, is guaranteed by the very definition of the... 

In this chapter, we will give a brief overview for the experimental findings and the main theoretical approaches for the ion effects in RNA and DNA thermal stabilities. We will then describe the TBI theory. We will focus on both the theoretical formalism and the practical applications of the theory. Our aim here is to provide sufficient detail so that all the major issues in the theoretical derivations and numerical computations can be clearly understood and readily followed. 

The expression for the nuclear shielding tensor a a does not depend on any particular value of ha- Though the spin magnetic moment of the nucleus must not be zero in order for magnetic resonance to occur at all in an experiment, in the theoretical formalism the nuclear shielding can be calculated anyway. In consequence, an explicit expression for <7 depends only on the position of the... 

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