  Let $L:=-\Delta+V$ be the Schr\"{o}dinger operator on $\mathbb{R}^n$ with
$n\geq 3$, where $V$ is a non-negative potential which belongs to certain
reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In
this article, the authors obtain the quantitative weighted boundedness of
Littlewood--Paley functions $g_L$, $S_L$ and $g_{L,\,\lambda}^\ast$, associated
to $L$, on weighted Lebesgue spaces $L^p(w)$, where $w$ belongs to the class of
Muckenhoupt $A_p$ weights adapted to $L$.
