Standing-wave packet

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

Illustration of the Heisenberg uncertainty principle for a wave packet, (a) A standing wave having Ap 0 has Ax oo (b) the superposition of standing waves having a nonzero but finite Ap has an intermediate vaiue of Ax as the wave packet becomes defined in space and (c) the superposition of standing waves where Ap oo causes Ax 0 and iocalizes the wave packet. 

It is important to emphasize that, in the above examples, knowledge of the PES was not required for the optimization process. The adaptive-control learning algorithm explores the available phase space and optimizes the evolution of the wave packet on the excited state PES without any prior knowledge of the surface. Thus, the intrinsic information about the excited-state dynamics of these polyatomic systems remains concealed in the detailed shape and phase of the optimized pulse. Inevitably, however, scientific curiosity, together with a desire to imder-stand how chemical reactions can be controlled, has led to pioneering studies that aim to identify the underlying rules and rationale that lead to a particular pulse shape or phase relationship that produces the optimum yield. 

As in wave mechanics, the simulation of chemical phenomena by number theory is characterized by the appearance of integers, in this case associated with chemical structures and transformations. An obvious conclusion is that the elementary units of matter should be viewed as wave structures rather than point particles, which is consistent with the first appearance of matter in curved space-time. Even 3D wave packets behave in a manner convincingly like ponderable matter and rationalize the equivalence of mass and energy in a natural way. There is no compelling reason why this simple model should be concealed with the notion of wave/particle duality and more so on realizing that the wave-like space-time distortions are strictly 4D structures. In response to environmental pressure, an electronic wave packet can shrink to the effective size of an elementary particle or increase to enfold a proton as a spherical standing wave. 

Interacting elementary wave packets are expected to coalesce into larger wave packets. All extranuclear electrons on an atom therefore together constitute a single spherical standing wave with internal structure, commensurate with a logarithmic optimization pattern. In the activated valence state, the central core of the wave packet is compressed into a miniscule sphere, compared to the valence shell which dominates the extranuclear space up to the ionization radius. 

These ionization radii, which have been shown [19] to underpin the electronegativity concept, have recently been derived by an extremely simple and more reliable simulation of atomic structure as a standing electronic wave packet [6]. This simulation, which is free of the errors of approximation that affect the HF simulation of small atoms, has produced a more reliable set of ionization radii, suitable for direct prediction of interatomic distance in general pairwise interaction within bonds of any order. The procedure is outlined in the next section. 

In a very strong standing light wave, an initially broad electron wave packet will be split into two parts, each being the envelope of many Bragg maxima, with the angle 20 between the split beams given by... 

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