Algebraic resonance quantization of coupled anharmonic oscillators

Abstract

A method called algebraic resonance quantization (ARQ) is presented for highly excited multidimensional systems. This approach, based on the Heisenberg form of the correspondence principle, is a fully quantum mechanical matrix method. At the same time, it uses modern nonlinear classical mechanics to greatly simplify the Hamiltonian matrix. For a model system of coupled Morse oscillators, a nonlinear resonance analysis shows that the Hamiltonian matrix is dominated by a few leading terms. This leads to an effective truncated sparse matrix whose diagonalization yields eigenvalues in excellent agreement with the exact values, even high in the chaotic regime. A new finding is that quantum couplings corresponding to rapidly oscillating, nonresonant terms can be important, and not just the higher-order resonant terms. The generalization of ARQ via a numerical semiclassical technique to many-dimensional systems with arbitrary couplings is outlined. The applicability of contemporary vector methods from quantum chemistry to ARQ of high vibrational levels is considered. The feasibility of sparse matrix ARQ methods for fitting spectra in the highly chaotic, ‘‘unassignable’’ regime is discussed.