Usually one can only resort to experience and classical physics to answer these questions. Next, one doesn't know why there are the de Broglie relations for momentum and energy and why the nonrelativistic energy-momentum relation is E=p2/2m.
Indeed, when Schrödinger originally invented his equation, he was also puzzled by the inevitable appearance of the imaginary unit 'i' in the equation. First, even if the behavior of microscopic particles likes wave and thus a wave function is needed to describe them, it is unclear why the wave function must assume a complex form.
There are at least two mysteries in such a heuristic derivation. The free Schrödinger equation in quantum mechanics is usually derived in textbooks by analogy and correspondence with classical mechanics.
