Nonrelativistic Fundamental Quantum and Classical Wave Equations

The irreducible representations of the extended Galilean group are used to derive infinite sets of symmetric and asymmetric second-order differential equations with constant coeffcients. All derived equations are local and their Lagrangians exist. It is shown that the asymmetric equations are Galilean invariant but the symmetric ones are not. By specifying quantum and classical physical settings, the constants in the equations are determined and the fundamental wave equations, including the Schr\"odinger, Schr\"odinger-like and new asymmetric equations, are obtained; the derived wave equation is non-fundamental. Formulation of wave theories based on the fundamental and non-fundamental wave equaions is considered, and physical implications of these theories on the wave description are discussed.