In $\mathbb R^n\times \mathbb R$ we consider the Schr\"odinger equation \begin{equation}\tag{$\mathcal E_1$} i\partial_t u (x,t)+{\rm H}_{k,\varsigma} u(x,t)=F (x,t) \end{equation} with given boundary values on $\mathbb R^n.$ Here ${\rm H}_{k,\varsigma}=(\Vert x\Vert^{2-\varsigma}\Delta_k-\Vert x\Vert^\varsigma)/\varsigma$ is a differential-difference operator on $\mathbb R^n,$ where $k$ is a multiplicity function for the Dunkl Laplacian $\Delta_k,$ and $\varsigma$ is either $1$ or $2.$ In the $k\equiv 0$ case, ${\rm H}_{0,1}$ is the Laguerre operator and ${\rm H}_{0,2}$ is the Hermite operator. In this paper we obtain Strichartz estimates for the Schr\"odinger equation $(\mathcal E_1)$. We then prove that Strichartz estimates for the Schr\"odinger equation \begin{equation}\tag{$\mathcal E_2$} i \partial_t \psi(x,t)+(1/\varsigma) \Vert x\Vert^{2-\varsigma} \Delta_k \psi(x,t)=F(x,t) \end{equation} can be obtained from those for $(\mathcal E_1).$ With the specializations $k\equiv 0$ and $\varsigma=2$, the equation $(\mathcal E_2)$ reduces to the Euclidean Schr\"odinger equation, while the case where $k\equiv 0$ and $\varsigma=1$ is already new.