  We discuss a new relation between the low lying Schroedinger wave function of
a particle in a one-dimentional potential V and the solution of the
corresponding Hamilton-Jacobi equation with -V as its potential. The function V
is $\geq 0$, and can have several minina (V=0). We assume the problem to be
characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller
quantum tunneling parameter $\epsilon$ between these different minima.
Expanding either the wave function or its energy as a formal double power
series in $g^{-1}$ and $\epsilon$, we show how the coefficients of
$g^{-m}\epsilon^n$ in such an expansion can be expressed in terms of definite
integrals, with leading order term determined by the classical solution of the
Hamilton-Jacobi equation. A detailed analysis is given for the particular
example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.
