  Recent cosmological observations indicate that the reionized universe may
have started at around z=6, where a significant suppression around Ly alpha has
been observed from the neutral intergalactic medium. The associated neutral
hydrogen column density is expected to exceed 10^{21}cm^{-2}},where it is very
important to use the accurate scattering cross section known as the
Kramers-Heisenberg formula that is obtained from the fully quantum mechanical
time-dependent second order perturbation theory. We present the
Kramers-Heisenberg formula and compare it with the formula introduced in a
heuristic way by Peebles (1993) treating the hydrogen atom as a two-level atom,
from which we find a devitaion by a factor of two in the red wing region far
from the line center. Adopting simple cosmological models, we compute the
Gunn-Peterson optical depths and the trough profiles. Our results are compared
with the works performed by Madau & Rees (2000), who adopted the cross section
introduced by Peebles. We find deviations up to 5 per cent in the Gunn-Peterson
transmission coefficient for an accelerated expanding universe in the red
off-resonance wing part with the rest wavelength Delta_lambda~10{AA}.
