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A Suggested Generalization of Born

G.R.Dixon, 4/21/2007

With regard to points where Y=0, there appear to be at least two possibilities: (1) the particle will interact with its environment if it is there, but it is never there; (2) the particle will not interact with its environment if it is there, but it is sometimes there.

For example, a double slit experiment might typify possibility 1. The bands of brightness occur where the waves from the slits interfere constructively. Practically all of the particles go there. And virtually none of the particles go where there is total darkness. Let us refer to such cases as "transverse interference."

A second type of interference occurs when a single wave is reflected and interferes with itself. At a perfect reflector there is full phase reversal and thus a node. Yet logically the particles must be at the reflector in order to be turned around. (Similar remarks apply at other standing wave nodes.) In such cases possibility 2 seems to be the correct explanation. We might refer to these cases as "longitudinal interference."

Let us abbreviate the two Y=0 possibilities in the following way: (1) not exists and is interactive; (2) exists and is not interactive. Since logical "and" plays a role, we might say that Y is the product of two other complex numbers, say cE (where subscript "E" indicates "exists") and cI (where subscript "I" indicates "interacts"):

Y=cE cI (1)

In cases of transverse interference the subject particles would interact anywhere but "choose" not to go everywhere. It is cE that varies from point to point and modulates Y. In the case of longitudinal interference the subject particles go everywhere but choose not to interact everywhere. It is cI that varies from point to point and modulates Y.

In a system such as the Hydrogen atom the orbiting electron may occupy any of many modes, ranging from linear oscillation through the nucleus to circular orbits around it. Oscillations through the nucleus would be accompanied by longitudinal interference, and circular orbits by transverse interference. (With regard to the latter, it should be borne in mind that a Born volume differential need not be cubical; it might be a spherical shell. cI might be single-valued at all R (shell radii) in such cases, whereas cE is nonzero only for discrete values of R.)

In any case the majority of H electron modes would constitute a combination of longitudinal and transverse interference. Assuming the "E" and "I" components add, we might expect to have in such hybrid cases

Y = (cE(L) + cE(T))( cI(L) + cI(T)). (2)