  The efficacy and accuracy of Bogomolny's method of the quantum surface of
section is evaluated by applying it to the quantization of the motion of a
particle in a smooth 2-D potential. This method defines a transfer operator T
in terms of classical trajectories of one Poincar\'e crossing; knowledge of T
provides information about the eigenstates of the quantum system. By using a
more robust quantization criterion than the one proposed by Bogomolny, we are
able to locate more than five hundred quantum states in both the regular and
the chaotic regimes--in most cases unambiguously--and see no reason that the
spectra could not be continued indefinitely. The errors of the predictions are
comparable in the two regimes, and roughly constant for increasing excitation,
but grow as a fraction of the (shrinking) mean level spacing. We also show
computed surface of section wavefunctions, and present other theoretical and
practical results related to the technique.
