With the electron-phonon scattering operator given in (classical) phase space scattering is described by Fermi's golden rule. In Chapter 9 we will include this classical description for scattering in the quantum Wigner Equation 8.1 to solve this model by Monte Carlo methods.
The fundamental reason, why this model cannot be adequately formulated within the QTBM, is that the corresponding equation for the density matrix cannot be separated, that is, there is no corresponding single particle Schrödinger equation. This model can only be formulated with the use of densities.
What we need technically is a method to describe a single particle (wave function, pure state) in contact with a phonon bath (density, mixed state) within the QTBM. Formulation of such models for scattering is the topic of the next subsections.