I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. JamalS recommended the Fourier transform, which is why I want to quote his example:

Example

As an example, consider the case $V(x,t)=\delta(t)$, in which case the Schrödinger equation becomes,

$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + \delta(t)\psi$$

We can take the Fourier transform with respect to $t$, rather than $x$, to enter angular frequency space:

$$-\hbar\omega \, \Psi(\omega,x)=-\frac{\hbar^2}{2m}\Psi''(\omega,x) + \psi(0,x)$$

which, if the initial conditions are known, is a potentially simple second order differential equation, which one can then apply the inverse Fourier transform to the solution.

Now, my question would be: What are meaningful initial conditions for this ODE? I mean, what you probably want to look at is how a wavefunction $\Psi(t=0,x)$ propagates in time? So how do you set up meaningful initial conditions for this Fourier-transformed Schrödinger equation?