  We investigate finite-size effects in quantum systems at first-order quantum
transitions. For this purpose we consider the one-dimensional q-state Potts
models which undergo a first-order quantum transition for any q>4, separating
the quantum disordered and ordered phases with a discontinuity in the energy
density of the ground state. The low-energy properties around the transition
show finite-size scaling, described by general scaling ansatzes with respect to
appropriate scaling variables. The size dependence of the scaling variables
presents a particular sensitiveness to boundary conditions, which may be
considered as a peculiar feature of first-order quantum transitions.
