Electron-phonon interactions and the states of a crystal electron in an external electric field

The effect of the electron-phonon interactions on the states of a single tightly bound electron in a one-dimensional crystal in an external electric field F is discussed with the help of a master equation for the occupation probabilities of electron states. It is found that the occupation probabilities of the Stark ladder states usually do not provide a good description of the state of the system despite the fact that these were eigenstates in the absence of the interactions. A good description is possible in terms of the occupation probabilities of time-dependent band states with $\text{k(t) = k}{}_{\mathrm{o}}\text{+ eFt/}\text{h}\text{̵}$ since these probabilities satisfy a markoffian master equation.