In this paper, we discuss in brief the most common wave equations in quantum mechanics and some recent development in wave mechanics. We also present two new quantum wave mechanics equations based on the Compton momentum. We also question the idea that energy and mass are scalars, and we claim they are vectors instead. We have good reasons to think that the standard momentum is a mathematical derivative of the more fundamental Compton momentum. This will hopefully simplify interpretations of quantum mechanics significantly; our new relativistic wave equations look promising, but need further investigation into what they predict. This way of looking at quantum mechanics in new light is not in conflict with existing equations, but they are supplemental to the collection of existing wave equations. We prove mathematically that if one satisfies our new relativistic energy Compton momentum relation, one also satisfies the standard relativistic energy momentum relation automatically. They are two sides of the same coin, where the relations to the Compton wavelength likely represent the deeper reality, so we have reasons to think our new wave mechanics addresses a deeper level of understanding than the existing conception. We also look at our new wave equation in relation to hydrogen-like atoms; we follow the “standard approach” used for the Schro ̈dinger equation of putting it in polar coordinate form and, by change of variables, finding three ODEs and their solutions. We give a table summary of our new ODEs and their solutions compared to the well-known solutions of the Schro ̈dinger equation.