The improvement wave equations of relativistic and non-relativistic quantum mechanics

In this work, we follow the idea of the De Broglie's matter waves and the analogous method that Schr\"{o}dinger founded wave equation, but we apply the more essential Hamilton principle instead of the minimum action principle of Jacobi which was used in setting up Schr\"{o}dinger wave equation. Thus, we obtain a novel non-relativistic wave equation which is different from the Schr\"{o}dinger equation, and relativistic wave equation including free and non-free particle. In addition, we get the spin 1/2 particle wave equation in potential field.