A new approach to the determination of good action-angle variables for coupled oscillator systems

Abstract

A theory of action-angle variables for coupled oscillator systems is developed which involves solving the Schrödinger equation using a basis of WKB eigenfunctions, then using the logarithm of the resulting wavefunction to define the generator for the canonical transformation which determines the action-angle variables. This theory is based on the marriage between Miller's method for solving the Hamilton-Jacobi equation using the logarithm of the generating function, and the Ratner-Buch-Gerber method for solving the Schrödinger equation using WKB basis functions. A perturbation-theory analysis of this theory indicates that the semiclassical eigenvalues and canonical transformations obtained from it should become identical to their exact classical counterparts in the limit of large actions for each vibrational mode. Two methods for systematically improving the theory for the lower eigenstates are also proposed. Numerical applications of the theory are presented for two systems, the Morse oscillator and the Henon-Heiles two-mode hamiltonian. The resulting semiclassical eigenvalues are in excellent agreement with their exact quantum counterparts, with the magnitude of the error roughly independent of the energy of the eigenstate. Analogous good agreement is found in comparing the approximate and exact classical canonical transformations. In particular, for the Morse oscillator, good results are obtained for certain higher energy states where second-order classical perturbation theory makes serious errors. Other information examined includes surfaces of section for the Henon-Heiles system (comparing the analytical functions obtained from the present theory with results based on exact trajectory calculations) and vibrational distributions chosen to simulate trajectory calculations (using the present theory to determine bin boundaries for a histogram calculation). Again, the comparison in each case with accurate results is excellent, with maximum errors in action calculations of 0.02 h, and in angle calculations of 0.01 rad.