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While it is true that Schrödinger’s argument was not rigorous (it was rather a “sleepwalker” argument 1 ), all the mathematically “forbidden” steps he took ultimately lead him to his famous equation (6.3). But it all worked so well, because what he was discovering, using rudimentary and awkward mathematical methods, was a property of pure mathematics. He in fact discovered the metaplectic representation of the symplectic group
chapter 6 in The Principles of Newtonian and Quantum Mechanics by M. Gosson