  We rigorously derive an effective quantum mechanical Hamiltonian from N=4
gauge theory in the BMN limit. Its eigenvalues yield the exact one-loop
anomalous dimensions of scalar two-impurity BMN operators for all genera. It is
demonstrated that this reformulation vastly simplifies computations. E.g. the
known anomalous dimension formula for genus one is reproduced through a
one-line calculation. We also efficiently evaluate the genus two correction,
finding a non-vanishing result. We comment on multi-trace two-impurity
operators and we conjecture that our quantum-mechanical reformulation could be
extended to higher quantum loops and more impurities.
