# Quantum field theory meson scattering calculation (scalar yukawa theory)

Please see this question for a clear background of the notation I use. My issue is that I want to use Wick's theorem to calculate the amplitude of meson $$\psi(p_1)\psi(p_2)\rightarrow\psi(p_1')\psi(p_2')$$ scattering. I can quickly get to the point where I need to evaluate:

$$\langle p_1',p_2'|:\psi_1^\dagger\psi_1\psi_2^\dagger\psi_2:|p_1,p_2\rangle$$

In my notes, the next step is

$$\langle p_1',p_2'|:\psi_1^\dagger\psi_1\psi_2^\dagger\psi_2:|p_1,p_2\rangle=\langle p_1',p_2'|\psi_1^\dagger\psi_2^\dagger|0\rangle\langle0|\psi_1\psi_2|p_1,p_2\rangle$$

where he seems to have changed orders of the fields and inserted $$|0\rangle\langle0|$$, $$|0\rangle$$ being the vacuum of the free theory. Can anyone explain what this particular step is? From thereafter the derivation is pretty clear to me

Expand the normal order: $$:\psi^\dagger_1\psi_1\psi^\dagger_2\psi_2:=\psi^\dagger_1\psi^\dagger_2\psi_1\psi_2.$$ (creator operator before annihilation operator). Then insert unit operator $I=|0\rangle\langle0|$ between them. Meaning of this expression is that, the total amplitude include two amplitudes. One is for annihilating initial particles of momentums $p_1, p_2$ to vacuum state, and one is for creating final particle of momentum $p'_1,p'_2$ from vacuum state. Due to the rule of probability multiplication, this must be a product.