Sub-Quantum Physics 1: Method of Fits Applied to the Hydrogen Atom
Keywords: quantum, hydrogen, causality, aether
This article is part of a program to explain the quantum uncertainty and the quantum wave function as envelopes of motion resulting from bombardment of a quantum particle by the surrounding medium or aether. The program sees quantum position probabilities as superpositions of position probabilities of classical orbits. This ?method of fits,? a method successful for the ground state of the oscillator, is applied here to states of the hydrogen atom. 1S and 2P states are studied in detail. The Schr?dinger solution for the 1S state is exactly fit by the method. The Schr?dinger solution for the 2P state is seen as a summary, but not an exact description, of a true dynamical state of motion in superposed coaxial ellipsoids. The general mathematical identity for the method of fits for hydrogen states shows the quantum position probabilities to be represented as superpositions of position probabilities of collections of segments of classical orbits. For hydrogen the classical orbits are ellipses, which are here grouped into ellipse families, each family consisting of ellipses all of which have the same major axis, the same energy, the same period, and the same action integral around the orbit. Ellipses in the family differ in their minor axes (which are proportional to their angular momenta). Various collections of segments of an ellipse family can be constructed by distinct choices of functions for weighting the minor axes and the completeness of ellipse orbit angular segments. The 2-dimensional ?standard? segment collection, in which all minor axes are weighted equally and all ellipses are complete, is shown to have a radial position probability density that is directly proportional to the radius, out to a distance equal to the major axis as the extreme radius. The 3-D radial probability function formed from the 2-D standard collection exactly fits the Schr?dinger 1S state. A mathematical identity is derived showing how the Schr?dinger radial probability function for any hydrogen state can be represented as a superposition of family segment collection probability profiles. Transitions between families are assumed to result from impulses from the aether, a theory of which will be required to predict the segment collection probabilities.