Fourier transform of positive momentum states only?

Suppose I have the momentum-space eigenstates of a system $psi_n(p)$. I write the time-evolved momentum states as $Psi(p,t) = sum c_npsi_n(p)e^{-i E_n t}$ where $E_n$ are the corresponding energy eigenvalues.

My Goal: To construct a position-space solution (not general) of the Schrödinger equation, formed entirely of positive momentum states.

Question:
My naïve guess was to take the Fourier transform over only positive momenta, i.e.
$$Phi(x,t) = int_{-infty}^infty Theta(p) Psi(p,t)e^{i p x} dp$$
where $Theta(p)$ is The Heaviside Step function.

But I could not show (probably isn’t true) that such a function would indeed be a solution to the Time-Dependent SE (which is satisfied by $Psi(p,t)$ in momentum-space).

Is there any method/idea to achieve the aforementioned ‘goal’?