  Let $X$ be a smooth quartic hypersurface in $\mathbb{P}^3$. By the
Brill-Noether theory of curves on K3 surfaces, if a rank 2 aCM bundle on $X$ is
globally generated, then it is the Lazarsfeld-Mukai bundle $E_{C,Z}$ associated
with a smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$. In this
paper, we will focus on the classification of such bundles on $X$ to
investigate aCM bundles of rank 2 on $X$. Concretely, we will give a necessary
condition for a rank 2 vector bundle of type $E_{C,Z}$ to be indecomposable
initialized and aCM, in the case where the class of $C$ in Pic($X$) is
contained in the sublattice of rank 2 generated by the hyperplane class of $X$
and a non-trivial initialized aCM line bundle on $X$.
