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.


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