Sin(omega t + psi ) integration
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Answer:
The short answer is, quantum mechanics only works if the wave phase is in the exponential form, with a complex function. If we go back to Hamilton, he actually developed a wave equation for classical mechanics. That got nowhere in particular because superficially it appears to avoid Newton’s first law. If we then retreat a square to the Hamilton-Jacobi equation, we find the main variable is the action S, which in turn is the time integral of the Lagrangian. Now, if we take the route to Hamilton’s waves, Schrödinger would note that while they did not make any sense in what they already knew about the quantum world, it was obvious that one route to make progress was to quantize the action, following Planck. (The fact the Bohr quantised angular momentum actually is not very helpful in seeing this. Angular momentum is dimensionally equivalent to action, but it is conceptually different because it is constant and a wave evolves.) Action evolves, the phase of a wave evolves, so it became obvious to Schrödinger that he should quantise the phase of the wave function, and since the phase must be a number (check your exponent), and because after a bit of algebra with Hamilton’s waves it became clear that the phase of the quantum wave had to be in the exponential form, he tried putting the phase as exp(2πiS/h), and after doing that, it is just a bit more algebra and you get the Schrödinger equation. So the answer to your question would appear to be that he knew he had to use action somewhere to properly quantise his waves, and using the algebra that Hamilton had started basically sent him in the exponential form.