I asked in this thread Time-dependet Schrödinger equation how to solve the Time-dependent Schrödinger equation. One of JamalS' recommendations was 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? You don't need to refer to this particular ODE(with this potential). My question is rather: When you solve this ODE, what are appropriate initial/boundary conditions for this Fourier transformed ODE, cause this is were my imagination fails.
If anything is unclear, please let me know.
Best Answer
Working in the frequency space helps simplify the differential equation you need to solve. Now it should be possible to find a bunch of solutions to the new differential equation. However, in the end what you want to solve is still the time-dependent one. So you need to come back to the initial or boundary conditions of the original time-dependent equation to fix the uncertainty. To be more specific, you can try to build the time-dependent wave function with those solutions you obtained. Certainly there will be unknown coefficients to be determined at the last step.