The correspondence of the previously given relativistic quantum mechanics of spinless or spin 1/2 particle with the quantum nonrelativistic and the classical relativistic mechanics is proved. In the Galilean quantum case the appearance of the familiar transformation phase factor is explained and the origin of the mass superselection rule indicated. In the relativistic classical case the equation for trajectories and the covariant equation of motion for spin are obtained. The conditions are found under which the usual theory of relativistic wave equations is of relevance (in some particular reference frame).
We show that the Schr\"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr\"odinger equation.