Model Systems and Approximate Constants of Motion

Abstract

We present a framework to quantify the extent to which an approximate Hamiltonian is a suitable model for a real Hamiltonian, based on the degree of stability of the approximate constants of motion that are exact constants in the model. By observing the evolution under the real Hamiltonian of packets prepared initially as eigenstates of the model Hamiltonian, we are able to define quantitative criteria for the quality 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 evaluating alternative starting points for perturbational and variational calculations, and for extracting physical insights from elaborate calculations of real systems. The use of the analysis is illustrated with examples of a one-dimensional Morse oscillator approximated by a harmonic oscillator and by another Morse oscillator, and then by a less trivial system, an anharmonic, nonseparable two-dimensiqnal oscillator, specifically a Henon-Heiles potential modified with a fourth-order term to keep all states bound. The higher the angular momentum within any given band, the better the angular momentum is conserved. The square of the angular momentum is less well conserved than the angular momentum itself. I. Introduction The concept of the “approximate constant of the motion” has never been given a precise, useful, quantitative meaning. Undoubtedly, this is in no small part because the approximate nature of the concept does not beg for rigor; we all have an intuitive idea of what the term connotes, and for most of us this is sufficient. By asking “How approximate?”, however, we can gain insight into the applicability of various trial systems as models for a real system. This is especially important to the researcher who makes extensive use of perturbational or variational techniques. The choice of basis set in perturbational or variational calculations is intimately tied to this concept. The choice of basis set implies a choiceof model system, each of which has its own set of constants of the motion. The extent to which these constants are maintained under conditions of the true Hamiltonian determine the suitability of the model system for emulating the true one. Manifestations of this success are the rate of convergence and quality of convergence of the representation. The earliest allusion that we could find to an approximate constant of the motion was due to van Vleck, when he wrote in 1932 on the quantization of electronic angular momentum in diatomic molecules.’ More recently, the idea has been used in the context of chaotic dynamics in molecules.2 Additionally, a method has been developed for the determination of exact and approximate constants of the motion for systems whose Hamiltonians can be expressed as polynomials in positions and momenta.3 We define an approximate constant of the motio! to be an observable X associated with an operator A such that A does not commute with the exact Hamiltonian, S, but does commute with SO, an approximation of the exact Hamiltonian. To be an approximateconstant of the motion, it must also satisfy one more condition: fluctuations of X about its mean value should be smaller than the spacing of the levels of the quantum number of the approximate Hamiltonian. Quantifying this and other useful criteria is the goal of this study.