  We use the dispersive properties of the linear Schr\"{o}dinger equation to
prove local well-posedness results for the Boltzmann equation and the related
Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$.
The proofs are based on the use of the (inverse) Wigner transform along with
the spacetime Fourier transform. The norms for the initial data $f_0$ are
weighted versions of the Sobolev spaces $L^2_v H^\alpha_x$ with $\alpha \in
\left( \frac{d-1}{2},\infty\right)$. Our main results are local well-posedness
for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as
well as local well-posedness for the Boltzmann hierarchy for cutoff Maxwell
molecules (but not hard spheres); the latter result holds without any
factorization assumption for the initial data.
