  The Poincar\'e symmetry can be contracted in two ways to yield the Galilei
symmetry and the Carroll symmetry. The Schr\"odinger equation exhibits the
Galilei symmetry and is a fundamental equation in Galilean quantum mechanics.
However, the question remains: what is the quantum equation that corresponds to
the Carroll symmetry? In this paper, we derive a novel equation, called the
Carroll-Schr\"odinger equation, which describes the quantum dynamics in the
Carrollian framework. We also introduce the Carroll-Schr\"odinger algebra,
which is a conformal extension of the centrally extended Carroll algebra, and
show that it is the symmetry algebra of the Carroll-Schr\"odinger equation in
two dimensions. We generalize our results to arbitrary dimensions and discuss
some possible applications and extensions of our work.
