  We continue our previous work [Ling-Bing He, Xuguang Lu and Mario Pulvirenti,
Comm. Math. Phys., 386(2021), no. 1, 143223.] on the limit of the spatially
homogeneous quantum Boltzmann equation as the Planck constant $\epsilon$ tends
to zero, also known as the semi-classical limit. For general interaction
potential, we prove the following: (i). The spatially homogeneous quantum
Boltzmann equations are locally well-posed in some weighted Sobolev spaces with
quantitative estimates uniformly in $\epsilon$. (ii). The semi-classical limit
can be further described by the following asymptotic expansion formula: $$
f^\epsilon(t,v)=f_L(t,v)+O(\epsilon^{\vartheta}).$$ This holds locally in time
in Sobolev spaces. Here $f^\epsilon$ and $f_L$ are solutions to the quantum
Boltzmann equation and the Fokker-Planck-Landau equation with the same initial
data.The convergent rate $0<\vartheta \leq 1$ depends on the integrability of
the Fourier transform of the particle interaction potential. Our new
ingredients lie in a detailed analysis of the Uehling-Uhlenbeck operator from
both angular cutoff and non-cutoff perspectives.
