# Invariant set under the flow defined by Schroedinger equations

I have to show that the set of functions of the form $$\psi(x,t)=c(t)^{-1}e^{\frac{-(x-q(t))^2}{2c(t)^2}}e^{ip(t)x}\hspace{1cm}c(t),p(t),q(t)\in\mathbb{R}$$ is invariant (as set) under the flow defined by Schroedinger equation $$i\frac{\partial\psi}{\partial t}=-\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}$$ and I have to find the orbit $\{q(t),p(t)\}$ corresponding to the initial datum $\{q_0,p_0,c_0\}$.

(I would show the invariance using the Fourier Transform but I don't know if it is right.)