{'fraction_non_alphanumeric': 0.027950310559006212, 'fraction_numerical': 0.008281573498964804, 'mean_word_length': 5.081761006289308, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hydrogen-Atom Wavefunctions as Radiationless Modes

After the discovery of the electron by J.J. Thomson [1], it was natural to consider that electrons are constituents of atoms, but if electrons in an atom are small electric charges moving in circular orbits, they would emit electromagnetic radiation and the radius of the orbit would quickly collapse to zero. In 1903, Thomson pointed out [3, 4] that a uniform ring of electric charge, rotating about its symmetry axis at constant angular velocity, does not emit electromagnetic radiation although the moving charge is subject to a centripetal acceleration. This was the first discovery of a “classical radiationless mode”. Thomson was inspired by this result to propose his “plum pudding” model of an atom [7] in which the positive electric charge was spread out in a continuous fashion, perhaps around rings of negative electrons. Little survives of this model today, except that the quantum probability current,