# Functional integral (semiclassic formula)

If to calculate the propagator K(x,x') (vaccuum)for a theory so:

$$(i\hbar \frac{\partial}{\partial t}\Psi - H\Psi )K(x,x')=\delta (x-x')$$ (1)

we use the functional integral approach:

$$K(x,x')=<0|e^{iS[x]/\hbar }|0>$$

my question is, let's suppose we use the semiclassical WKB approach to calculate $$K_{WKB} (x,x')$$ my question is ¿does the classical propagator satisfies the Schöedinguer equation (1) or as an approximation i'd like to know if the semiclassical propagator satisfies a Hamilton-Jacobi type equation... thanks.