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On the Page no.102, Last Paragraph of Peskin and Schroeder,
...If we set up $|\phi_\mathcal{A}\phi_\mathcal{B}\rangle$ in the remote past, and then take the limit in which the wavepackets $\phi_i(\mathbf{k}_i)$ become concentrated about the definite momenta $\mathbf{p}_i$, this defines an in state..
What is the limit in which wave packets become concentrated about definite momenta? What did the book mean by this?
In the distant past, the interaction part of the Hamiltonian is zero, and hence the Hamiltonian is a free (non-interacting) Hamiltonian. This means that at $t=-\infty$, the eigenstates of the Hamiltonian have definite momenta. We take these to be our `in' states.
Imagine a scattering experiment, in which we fire two particles at each other. At the start of the experiment, these particles will be separated by a large distance, and can be taken to be free particles with definite momenta. There is no contribution to the Hamiltonian from their interaction. When they meet, they interact in some complicated way, and then at the end of the experiment, far separated again, they will once more be free. Which states the particles end up in (i.e., with what probability), is where the S matrix comes in.
