  We study finite-size effects at first-order quantum transitions (FOQTs). We
show that the low-energy properties show a finite-size scaling (FSS) behavior,
the relevant scaling variable being the ratio of the energy associated with the
perturbation driving the transition and the finite-size energy gap at the FOQT
point. The size dependence of the scaling variable is therefore essentially
determined by the size dependence of the gap at the transition, which in turn
depends on the boundary conditions. Our results have broad validity and, in
particular, apply to any FOQT characterized by the degeneracy and crossing of
the two lowest-energy states in the infinite-volume limit. In this case, a
phenomenological two-level theory provides exact expressions for the scaling
functions. Numerical results for the quantum Ising chain in transverse and
parallel magnetic fields support the FSS ansatzes.
