  Weak measurements are a subset of measurement processes in quantum mechanics
wherein the system which is being measured interacts very weakly with the
measuring apparatus. Measurement values of observables undergoing a weak
interaction and their amplification, are concepts that have sharpened our
understanding of interaction processes in quantum mechanics. Recent experiments
show that naturally occurring processes such as resonance fluorescence from
excited states of an atom can exhibit weak value amplification effect. In this
paper, we theoretically analyze the process of elastic resonance fluorescence
from a V-type three level atomic system, using the well known Weiskopff-Wigner
(W-W) theory of spontaneous emission. Within this theory, we show that, a weak
interaction regime can be identified and for suitable choices of initial and
final excited states, the mean scattering time between these states show an
amplification effect during interaction with the vacuum bath modes of the
electromagnetic field. We thus show for the first time that a system bath
interaction can show weak value amplification. Using our theory we reproduce
the published experimental results carried out in such a system. More
importantly, our theory can calculate scattering timescales in elastic
resonance scattering between multiple excited states of a single atom or
between common excited state configurations of interacting multi-atom systems.
