  We study the time behavior of wave functions involved in tunneling through a
smooth potential barrier in one dimension in the semiclassical limit. We
determine the leading order component of the wave function that tunnels. It is
exponentially small in $1/\hbar$. For a wide variety of incoming wave packets,
the leading order tunneling component is Gaussian for sufficiently small
$\hbar$. We prove this for both the large time asymptotics and for moderately
large values of the time variable.
