  The aim of this article is to relate the discrete quantum walk on
$\mathbb{Z}$ with the continuous Schr\"odinger operator on $\mathbb{R}$ in the
scattering problem. Each point of $\mathbb{Z}$ is associated with a barrier of
the potential, and the coin operator of the quantum walk is determined by the
transfer matrix between bases of WKB solutions on the classically allowed
regions of both sides of the barrier. This correspondence enables us to
represent each entry of the scattering matrix of the Schr\"odinger operator as
a countable sum of probability amplitudes associated with the paths of the
quantum walker. In particular, the barrier-top scattering corresponds to the
Hadamard walk in the semiclassical limit.
