Eliminating remnants of classical mechanics and the birth of the Schr\"odinger equation

We show that the Schr\"odinger equation emerges from the Hamilton-Jacobi equation for a specific choice of the amplitude $R$ of a wave $\psi\equiv R\exp[I S/\hbar]$ where $S$ is the classical action. This choice eliminates in the wave equation for $\psi$ all remnants of classical mechanics associated with $S$ but at the same time builds via the wave equation for $R$ a bridge to classical mechanics and to the de Broglie pilot wave theory.