Potential electromagnetic wave

Determination of ionization potentials of 89 molecules) 5) W.W. Belevary, "Interaction Between Electromagnetic Waves and Flames , Part 5 Sources of Ionization in Rocket Exhaust , NOTS 1708(1957) (Conf)... 

The accurate quantum mechanical first-principles description of all interactions within a transition-metal cluster represented as a collection of electrons and atomic nuclei is a prerequisite for understanding and predicting such properties. The standard semi-classical theory of the quantum mechanics of electrons and atomic nuclei interacting via electromagnetic waves, i.e., described by Maxwell electrodynamics, turns out to be the theory sufficient to describe all such interactions (21). In semi-classical theory, the motion of the elementary particles of chemistry, i.e., of electrons and nuclei, is described quantum mechanically, while their electromagnetic interactions are described by classical electric and magnetic fields, E and B, often represented in terms of the non-redundant four components of the 4-potential, namely the scalar potential and the vector potential A. 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

Therefore, questioning the physical significance of potential is not relevant here. The new formulation of Maxwell s equations [20-23], where potentials are directly coupled to fields clearly indicates that potentials, play a key role in particle behavior. To make a long story short, the difference in nature between potentials and fields stems from the fact that potentials relate to a state of equilibrium of stationary waves in the medium usually nonaccessible to an observer (except when potentials are used in a measurement process of the interferometric kind, at a given instant in time). Conversely, fields illustrate a nonequilibrium state of the medium as an observable progressive electromagnetic wave, since this wave induces the motion of material particles. 

The simplest example of the generation of energy from a pure gauge vacuum is to consider the case of an electromagnetic potential plane wave defined by... 

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

In terms of multipoles only the electric dipole is coupled to the electromagnetic field and so this approximation is termed the electric dipole approximation. It may appear strange that the electromagnetic field, which is transversal, in this approximation is given solely by the scalar potential. It must, however, be remembered that the scalar and vector potentials of (197) do not describe the electromagnetic wave as such. Rather, it models the interaction of the electromagnetic wave with the molecule [65]. 

The potentials for a uniform electric field corresponds to those of the electric dipole approximation, except that in the latter case the electric field is uniform only over the molecular volume and not over the entire space. Uniform fields extending over all space must be considered a mathematically convenient idealization they are allowed by Maxwell s equations, but not realizable experimentally. The same holds true for the source-free electromagnetic waves discussed above. It is also important to note that uniform fields do not form the static limit of an electromagnetic wave, in the limit % — 0 both the electric and magnetic fields go to zero. 

In this section, we are concerned with the canonical equations of the radiation field. We consider the fact that the electromagnetic wave is a transverse wave, and convert it into the form of Hamilton kinetic equations which are independent of the transformation parameter. In this process we will reach the conclusion that the radiation field is an ensemble of harmonic oscillators. During this process we will stress the concepts of vector potential and scalar potential. The equations of an electromagnetic wave in the vacuum are summarized as follows ... 

The general solution of these equations of the electromagnetic wave yields a rather complicated mathematical expression. So, by using the second equation of Eq. (1.60) and remembering the formula, V(Vx ) = 0 given in Table 1.3, we introduce the vector potential, A(x,y,z,t), which satisfies the following equations ... 

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