  We derive effective Hamiltonians for the fractional quantum Hall effect in
n=0 and n=1 Landau levels that account perturbatively for Landau level mixing
by electron-electron interactions. To second order in the ratio of
electron-electron interaction to cyclotron energy, Landau level mixing is
accounted for by constructing effective interaction Hamiltonians that include
two-body and three-body contributions characterized by Haldane
pseudopotentials. Our study builds upon previous treatments, using as a
stepping stone the observation that the effective Hamiltonian is fully
determined by the few-body problem with N=2 and N=3 electrons in the partially
filled Landau level. For the n=0 case we use a first quantization approach to
provide a compact and transparent derivation of the effective Hamiltonian which
captures a class of virtual processes omitted in earlier derivations of
Landau-level-mixing corrected Haldane pseudopotentials.
