The masses of S- and P-wave heavy-light mesons are computed in quenched QCD
using a classically and tadpole-improved action on anisotropic lattices. O
f particular interest are the splittings among P-wave states, which have no
t yet been resolved experimentally; even the ordering of these states conti
nues to be discussed in the literature. The present work leads to upper bou
nds for these splittings, and is suggestive, but not conclusive, about the
ordering.