The author considers the Schrödinger equation corresponding to the scattering metric on the Euclidean space and study the propagation of singularities for the solution in terms of the homogeneous wavefront set. Let \(X= S^n_+\) and \(x= z^{-1}\) for \(z\in\mathbb{R}^n\setminus\{0\}\) define a boundary defining function of \(X\) and a scattering metric on \(X\) which is expressed near the boundary as \(g= {dx^2\over x^4}+ {h\over x^2}\), where \(h\) is a Riemannian metric on \(\partial X\). Consider the Schrödinger operator \[ H= -{1\over 2\sqrt{g}} \sum^n_{j,k=1} \partial_j g^{jk}\sqrt{g}\partial_k +V. \] Assume that \(V\) is a smooth real-valued function on \(\mathbb{R}^n\) and satisfies \(|\partial^\alpha V(z)|\leq C_\alpha\langle z\rangle^{\nu-|\alpha|}\), \(\alpha\in\mathbb{Z}^n_+\) uniformly in \(z\in\mathbb{R}^n\), where \({3\over 2}\leq\nu< 2\). Define the homogeneous wavefront set (respectively, usual wavefont set) as follows; for \(S'(\mathbb{R}^n)\) and \((z_0,\zeta_0)\in T^*\setminus\{0\}\) \((z_0,\zeta_0)\not\in HWF(u)\) (respectively, \(WF(u)\)), if there exists \(\phi\in C^\infty_0(\mathbb{R}^{2n})\) such that \(\phi(z_0,\zeta_0)\neq 0\) and that \(\|\phi^w(hz, hD)u\|_{L^2}= O(h^\infty)\) (respectively \(\|\phi^w(z, hD)u\|_{L^2}= O(h^\infty)\)). Let \(u_0\in L^2(\mathbb{R}^n)\) and assume \((z_0,\zeta_0)\in T^*\setminus\{0\}\) is backward nontrapping and denote by \(\omega_-\) a backward limiting direction. Then the author proved that there exists \(t_0> 0\) such that \((-t_0\omega_-, \omega_-)\not\in HWF(u_0)\), \((z_0,\zeta_0)\not\in WF(e^{-it_0 H}u_0)\).