Semiclassical quantization using Bogomolny's quantum surface of section

Abstract

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.