It is shown that the heuristic derivation of the free Schrödinger equation in quantum mechanics textbooks can be made more strict by resorting to spacetime translation invariance and relativistic invariance.
In this article, we will demonstrate that the key to unveil these mysteries is to analyze the origin of momentum and energy. This is unsatisfactory in logic because quantum mechanics is a more fundamental theory, of which classical mechanics is only an approximation. 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.
