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