A number-theoretic method using Gaussian integers finds all equal-energy-difference transition pairs in the Bohr model and shows cascades longer than three levels are impossible via Fermat's theorem.
Bohr ,\ 10.1080/14786441308634955 journal journal Philosophical Magazine \ volume 26 ,\ pages 1 ( year 1913 ) NoStop
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Indistinguishable fermions generate correlations in quantum networks impossible for bosons or distinguishable particles without additional communication, establishing fermions as fundamentally more nonlocal.
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The Schrödinger equation is derived assuming the wave function is a probability amplitude and using the de Broglie relations E = ħω and p = ħk for the associated wave.
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