  It is expected in physics that the homogeneous quantum Boltzmann equation
with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann
operator (neglecting effect of the statistics) for the weak coupled gases will
converge to the homogeneous Fokker-Planck-Landau equation as the Planck
constant $\hbar$ tends to zero. In this paper and the upcoming work
\cite{HLP2}, we will provide a mathematical justification on this
semi-classical limit. Key ingredients into the proofs are the new framework to
catch the {\it weak projection gradient}, which is motivated by Villani
\cite{V1} to identify the $H$-solution for Fokker-Planck-Landau equation, and
the symmetric structure inside the cubic terms of the collision operators.
