  We study the norm approximation to the Schr\"odinger dynamics of $N$ bosons
in $\mathbb{R}^3$ with an interaction potential of the form
$N^{3\beta-1}w(N^{\beta}(x-y))$. Assuming that in the initial state the
particles outside of the condensate form a quasi-free state with finite kinetic
energy, we show that in the large $N$ limit, the fluctuations around the
condensate can be effectively described using Bogoliubov approximation for all
$0\le \beta<1/2$. The range of $\beta$ is expected to be optimal for this large
class of initial states.
