The free Schrödinger equation is shown to be a consequence of space-time homogeneity in the non-relativistic domain. First, we show that space-time translation invariance of natural laws, which is a conseqiuence of space-time homogeneity, entails that the state of a free particle with definite momentum and energy assumes the plane wave form when the time evolution of the state is linear. Next, we show that the conservation of energy and momentum, which is a consequence of space-time translation invariance, may further determine the energy-momentum relation In the non-relativistic domain. These results then lead to the free Schrödinger equation naturally. The new integrated analysis may help understand the origin of the wave equations in quantum theory.

Schrödinger equation, wave equations, space-time homogeneity, space-time translation invariance, conservation of energy and momentum