  We study how the classical Hamilton's principal and characteristic functions
are generated from the solutions of the quantum Hamilton-Jacobi equation. While
in the classically forbidden regions these quantum quantities directly tend to
the classical ones, this is not the case in the allowed regions. There, the
limit is reached only if the quantum fluctuations are eliminated by means of
coarse-graining averages. Analogously, the classical Hamilton-Jacobi scheme
bringing to the motion's equations arises from a similar formal quantum
procedure.

Quantum Physics