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Inspired by Holstein's work on small polaron hopping, the evolution equationsof localized states and extended states in presence of atomic vibrations arederived for an amorphous semiconductor. The transition probabilities areobtained for four types of transitions: from one localized state to anotherlocalized state, from a localized state to an extended state, from an extendedstate to a localized state, and from one extended state to another extendedstate. At a temperature not too low, any process involving localized state isactivated. The computed mobility of the transitions between localized statesagrees with the observed `hopping mobility'. We suggest that the observed`drift mobility' originates from the transitions from localized states toextended states. Analysis of the transition probability from an extended stateto a localized state suggests that there exists a short-lifetime belt ofextended states inside conduction band or valence band. It agrees with the factthat photoluminescence lifetime decreases with frequency in a-Si/SiO$_{2}$quantum well while photoluminescence lifetime is not sensitive to frequency inc-Si/SiO$_{2}$ structure.

Phonon driven transport in amorphous semiconductors: Transition probabilities