We want to find what with length the center of a incoming wave packet appreciate variations. The scattering approach prefer to talk about time and momentum in detriment of space, but we can talk about space after.
The scatering state always can be written as $$ \psi^{(+)}g(t)=\int d\alpha\, e^{-iE\alpha t} g(\alpha) \phi_\alpha + \int d\alpha\,\int d\beta\, \frac{e^{-iE_\alpha t} g(\alpha)T_{\beta \alpha}}{E_\alpha -E_\beta + i\epsilon} \phi_\beta $$
Where $g(\alpha)$ represent a suitable superposition of free-states $\phi_\alpha$. Note that when we send $ t \rightarrow - \infty $ the state $\psi^{(\pm)}g(t)$ becomes $\int d\alpha\, e^{-iE\alpha t} g^{(\pm)}(\alpha) \phi_\alpha$ because the pole in $\frac{1}{E_\alpha -E_\beta \pm i\epsilon}$. The term $T_{\beta \alpha}$ is responsible for include interactions. If we have $H=H_0+V$ then $T_{\beta \alpha}=\langle \beta|V|\alpha^{+}\rangle$, where $$ |\alpha^{+}\rangle=|\alpha\rangle+\frac{V}{E_\alpha -H_0 \pm i\epsilon}|\alpha^{+}\rangle . $$ We can calculate the integral on $\alpha$ in the second term by simple analysis of poles contained in $T_{\beta \alpha}$ as function of complex $\alpha$. 592
This is because we can deforming the integral for a large semi circle in the upper or lower complex plane and the integral in this contour is damped by $\frac{e^{-iE_\alpha t}}{E_\alpha - E_\beta+i\epsilon}$. Then only the poles and cuts of $T_{\beta \alpha}$ in complex plane contribute to the integral. For positive times, in the lower, for negative times, in the upper.
The most close pole to the real axis define the typical energy and time of scattering.
For calculate the typical length is simple take the momentum and mass of the in-state and confront to the typical time. Typical time is the inverse of the distance of the pole in units of $\hbar$. This relation of energy and time is a result of the damping mechanism of $\frac{e^{-iE_\alpha t}}{E_\alpha - E_\beta+i\epsilon}$.
In terms of Green's Functions we can calculate this pole. We simply take the unperturbed Green's Function $ G_0(\alpha)= \frac{1}{E_\alpha-H_0 \pm \epsilon}$ and calculate the time of an in-particle stay unperturbed. 795