  In this paper, we study weight representations over the Schr{\"o}dinger Lie
algebra $\mathfrak{s}_n$ for any positive integer $n$. It turns out that the
algebra $\mathfrak{s}_n$ can be realized by polynomial differential operators.
Using this realization, we give a complete classification of irreducible weight
$\mathfrak{s}_n$-modules with finite dimensional weight spaces for any $n$. All
such modules can be clearly characterized by the tensor product of
$\mathfrak{so}_n$-modules, $\mathfrak{sl}_2$-modules and modules over the Weyl
algebra.
