  We consider a wave-function approach to the false vacuum decay with gravity
and present a new method to calculate the tunneling amplitude under the WKB
approximation. The result agrees with the one obtained by the Euclidean
path-integral method, but gives a much clearer interpretation of an instanton
(Euclidean bounce solution) that dominates the path integral. In particular,
our method is fully capable of dealing with the case of a thick wall with the
radius of the bubble comparable to the radius of the instanton, thus surpassing
the path-integral method whose use can be justified only in the thin-wall and
small bubble radius limit. The calculation is done by matching two WKB wave
functions, one with the final state and another with the initial state, with
the wave function in the region where the scale factor of the metric is
sufficiently small compared with the inverse of the typical energy scale of the
field potential at the tunneling. The relation of the boundary condition on our
wave function for the false vacuum decay with Hartle-Hawking's no-boundary
boundary condition and Vilenkin's tunneling boundary condition on the wave
function of the universe is also discussed.
