I found the Fourier transform at $t=0$ for the wave function of a wave packet (and it's inverse Fourier transform) :
$$\Psi(x,0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(k)e^{ikx}dk$$ $$\Phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Psi(x,0)e^{-ikx}dx$$
Now I want to know the same Fourier transform, but at a time $t\neq0$
What would be the formula ?
My guess is:
For the Fourier transform: $\Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(k)e^{i(kx-\omega t)}dk$
For the inverse Fourier transform: $\Phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Psi(x,t)e^{i(-kx-\omega t)}dx$
But I am really not sure