  We use the semi-classical approximation in perturbative scalar quantum
electrodynamics to calculate the quantum correction to the Larmor radiation
formula to first order in Planck's constant in the non-relativistic
approximation, choosing the initial state of the charged particle to be a
momentum eigenstate. We calculate this correction in two cases: in the first
case the charged particle is accelerated by a time-dependent but
space-independent vector potential whereas in the second case it is accelerated
by a time-independent vector potential which is a function of one spatial
coordinate. We find that the corrections in these two cases are different even
for a charged particle with the same classical motion. The correction in each
case turns out to be non-local in time in contrast to the classical
approximation.
