Density Matrix in Scattering Theory [Part 1]

The scattering matrix S, is defined by

(1)
\begin{align} |\psi_{out}\rangle = S|\psi_{in}\rangle \end{align}

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)
\begin{align} \rho_{in} = \sum_{i}W_{i}|\psi^{i}_{in}\rangle\langle \psi^{i}_{in}| \end{align}

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)
\begin{align} \rho'_{out} = S\rho_{in}S^{\dagger} = \sum_{i}W_{i}S|\psi^{i}_{in}\rangle\langle \psi^{i}_{in}|S^{\dagger} = \sum_{i}W_{i}|\psi^{i}_{ out}\rangle\langle \psi^{i}_{out}| \end{align}

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

(4)
\begin{equation} T = S - 1 \end{equation}

Now,

(5)
\begin{align} T|\psi_{in}\rangle = |\psi_{out}\rangle - |\psi_{in}\rangle \end{align}

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)
\begin{align} \rho_{out} = T\rho_{in}T^{\dagger} \end{align}
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