A Suggested Re-interpretation of Born
According to the classical paradigm of Maxwell and Lorentz the electromagnetic field and the electromagnetic force, experienced by a charged particle, are continuously distributed in space and time. Einsteinís explanation of the photoelectric effect suggested that interactions between the field and charged particles do not occur continuously in space and time, but rather occur impulsively with no interaction between impulses. Over macroscopic distances, and/or when an object has many charges, little is changed by such impulsive interactions. That is, any "jerkiness" attributable to the impulsive nature of the interactions is "smoothed out" over macroscopic space/time intervals, and large scale behavior remains predictable and measurable.
An important distinction can be recognized, however, in microscopic cases. For example, in the case of a particle bound to a microscopic volume by a central force, an impulse might occur on one side of the center of force, and the next one might not occur until the particle has traveled to the other side of the volume. Consequently, microscopic phenomena are more statistical than determinate.
A suggested stratagem was first hinted at by Born, who conjectured that the square of the wave function is proportional to the probability that a particle will (or could) be found in a given volume. Little direction was suggested, however, regarding how microscopic particles might be found in a given volume. It is suggested here that it is the impulsive interaction between a field and a bound particle that amounts to "finding" the particle. In other words, it is suggested that the squared solution of Schrödingerís equation is proportional to the probability that an impulsive interaction will occur at a given point. By "probability" it is meant that, in a very great number of interactions, the fraction occurring in a given volume is a statement of the probability of an interaction occurring in that volume.
By itself, knowledge about where an interaction might occur provides only partial information. However, a companion (Fourier transform) of the quantum wave function provides the probability that a particle will have a given momentum. It is suggested here that this refers to the momentum directly following an interaction. Together, the quantum wave function and its transform provide statistical information about the location and momentum of a microscopic particle in a central force field.