 Gaussian eigenstate for a quadratic potential

For $U(x)=k{x}^2$ and $k>0$, the ansatz $\psi =C{e}^{-\alpha x^2}$ solves the stationary Schrödinger equation whenIt has $\Delta x=1/(2\sqrt{\alpha })$ and $\Delta p=\hslash \sqrt{\alpha }$, so it saturates the Heisenberg bound.