MODEL SYSTEMS AND APPROXIMATE CONSTANTS OF MOTION

We present a framework to quantify the extent to which an approximate Hamiltonian is a suitable model for a real Hamiltonian, based on the d egree of stability of the approximate constants of motion that are exa ct constants in the model. By observing the evolution under the real H amiltonian of packets prepared initially as eigenstates of the model H amiltonian, we are able to define quantitative criteria for the qualit y of the approximation represented by the model. Quantitative measures emerge for the concepts of ''approximate constant of the motion'' and ''pretty good quantum number''. This approach is intended for evaluat ing alternative starting points for perturbational and variational cal culations, and for extracting physical insights from elaborate calcula tions of real systems. The use of the analysis is illustrated with exa mples of a one-dimensional Morse oscillator approximated by a harmonic oscillator and by another Morse oscillator, and then by a less trivia l system, an anharmonic, nonseparable two-dimensional oscillator, spec ifically a Henon-Heiles potential modified with a fourth-order term to keep all states bound. The higher the angular momentum within any giv en band, the better the angular momentum is conserved. The square of t he angular momentum is less well conserved than the angular momentum i tself.