This paper deals with the Schrödinger equation
$i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where
$ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data
$f$ satisfies
$\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where
${q}_{s} $ is the heat kernel associated to
$ \mathcal{L} . $ If in addition
$\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some
${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that
$u(\mathbf{z} , t; s)= 0$ for all
$s\in \mathbb{R} $ whenever
$\alpha \beta \lt { s}_{0}^{2} . $ This result holds true in the more general context of
$H$-type groups. We also prove an analogous result for the Grushin operator on
${ \mathbb{R} }^{n+ 1} . $