The scattering matrix S, is defined by

(1)The scattering operator S can the thought of as a black box that transforms the 'in' states to the 'out' states.

The initial state can be represented by the density matrix $\rho_{in}$ defined by

(2)where the scalar $W_{i}$ represents the weight (contribution) of the in state $|\psi^{i}_{in}\rangle$ in the initial state.

The density matrix $\rho'_{out}$ is given by

(3)This is the density matrix of the 'out' state describing the final particles. As we are usually interested in the *transition* between different states, we can extract the unit operator $1$ from S to define the *transition operator* $T$ by

Now,

(5)Clearly, T transforms the "in" state into the scattered state. Therefore, the interesting part of the density matrix, which contains information about the scattered states alone, is given by

(6)