Quantum Physics & Pythagorean Triples
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Dynamic Structure of
the Hydrogen Atom
The hydrogen atom electron wavefunctions of a bound electron are developed in detail. The wavefunctions are the product of the real energy eigenfunctions of the time independent Schrödinger equation times the unit phasor, a complex rotating vector in the complex plane. This produces the hydrogen atom wavefunction as a rotating phasor on the complex plane modulated by the real energy eigenfunction providing a probability density contribution at each of the instantaneous points along the wavefunction phasor. The real probability densities determined for the bound electron in the hydrogen atom is then the product of the full electron wavefunctions times their complex conjugates in accordance with Max Born’s rule (1925). These real probability densities give the bound electron in the hydrogen atom an oscillating physical structure contrary to the static physical structure found in the early literature where the real number characterization of the energy eigenfunctions was not taken into account in the determination of the full wavefunctions.
The Calculation of
All Pythagorean Triples
Pythagorean Triples have been of intense interest since discovered by the Pythagoreans in ancient Greece. Only a small number of Pythagorean Triples are known, but their actual numbers are without bounds. Setting out to find a general system to find any and all Pythagorean Triples in principal, it was realized that a search algorithm had to be invented. Named Staircase, it was governed by natural numbers extending from 1 to as large as practicality limits.
For the development of Staircase in Table SA, a sub-column with 41 natural numbers starting at 1 incrementing 1 unit per row goes down Table SA. The first compete column of several sub-columns of Staircase is labeled Step 1. As the first column of Staircase Search has 41 ever longer entries of trial short sides, the ever larger short side forces a longer trial long side and a longer trial hypotenuse. By the design of Staircase if the trial long side and trial hypotenuse are also integers, then a Pythagorean triangle is produced as well as a new Pythagorean triple. Step 1 has the following Pythagorean Triples: {3, 4, 5}, {5, 12, 13}, {7, 12, 13} and [9, 40, 41}. Step 2 starts with a value of short side of 2 in the column of natural numbers. All further columns are one step down from the previous column. The first triple in column Step 2 is {4, 3, 5}. The first composite triple is in Step 2: {10, 24, 26} = 2 {5, 12, 13}. The next step with triples is Step 8. The first triple in Step 8 is {12, 5, 13}. Some further steps have no triples.
Staircase was searched for patterns in each Step. Patterns in those triples without a common factor in a given Step were found in Staircase Steps 1, 2, 8, 9, and 18. Analysis of these results lead to the formation of three General Triples Sequences (GTS) which produce all Pythagorean Triples to infinity. A wide analysis of the GTS is followed by further wide extensions to solidify understanding.