# A relation for a scattering state using second quantization

Consider the scattering of a particle with momentum $$\textbf{k}$$ and energy $$\varepsilon$$ by any target. I saw in an article that, if $$\Psi_{\textbf{k}}^{(+)}$$ is the scattering state with outgoing boundary conditions, we can write it as

$$\Psi_{\textbf{k}}^{(+)} = \lim_{t \rightarrow - \infty } e^{-i\varepsilon t} \hat{c}^{\dagger}_{\textbf{k}}(t) \Phi,$$

where $$\Phi$$ is the state of the target isolated and $$\hat{c}^{\dagger}_{\textbf{k}}(t)$$ is a criation operator in the Heisemberg picture. But where is it from? It sounds intuitive, but I can't find the deductionn of this relation anywhere.