  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.
