  We derive analytic expansions for the finite-volume energies of
weakly-interacting two-particle systems, using the general relations between
scattering amplitudes and energies derived by L\"uscher and others. The
relations hold for ground and excited states with both zero and non-zero total
momentum in the finite-volume frame. A number of instructive aspects arise in
the derivation, including the role of accidental degeneracies and the
importance of defining a power-counting scheme in the expansions. The results
give intuition concerning the imprint of weakly-interacting systems on the
energy spectrum, while also providing a useful basis for the analogous results
concerning three-particle excited states, to appear.
