  Stimulated Raman scattering (SRS) has played an increasingly pivotal role in
chemistry and photonics. Recently, understanding of light-molecule interaction
during SRS was brought to a new quantitative level through the introduction of
stimulated Raman cross section, $\sigma_{SRS}$. Measurements of Raman-active
molecules have revealed interesting insights, and theoretical consideration has
suggested an Einstein-coefficient-like relation between $\sigma_{SRS}$ and the
commonly used spontaneous Raman cross sections, $\sigma_{Raman}$. However, the
theoretical underpinning of $\sigma_{SRS}$ is not known. Herein we provide a
full quantum mechanical treatment for $\sigma_{SRS}$, via both a semi-classical
method and a quantum electrodynamic (QED) method. The resulting formula
provides a rigorous theory to predict experimental outcome from first
principles, and unveils key physical factors rendering $\sigma_{SRS}$
inherently strong response. Through this formula, we also confirm the validity
of the Einstein-coefficient-like equation connecting $\sigma_{Raman}$ and
$\sigma_{SRS}$ reported earlier, and discuss the inherent symmetry between all
spontaneous and stimulated optical processes. Hence the present treatment shall
deepen the fundamental understanding of the molecular response during SRS, and
facilitate quantitative applications in various experiments.
