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Journal of Physics B Atomic and Molecular Physics

A modified correspondence principle for strongly coupled states

A P Clark

Published Jul 1, 1973• 12 citations
Publisher

Abstract

The one-dimensional form of the correspondence principle for strongly coupled states of Percival and Richards (1970) is modified to include, approximately, the perturbation of the bound particle orbit. The modification will only be significant when the expectation value of the interaction potential depends on the state of the bound system. The excitation of a linear harmonic oscillator by a time dependent potential q2F(t) and the collinear collision of an atom and an anharmonic diatomic molecule are considered as examples. The results obtained for both these systems are in good agreement with the exact quantum mechanical transition probabilities even for low order transitions.

DOI
10.1088/0022-3700/6/7/010

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The one-dimensional form of the correspondence principle for strongly coupled states of Percival and Richards (1970) is modified to include, approximately, the perturbation of the bound particle orbit. The modification will only be significant when the expectation value of the interaction potential depends on the state of the bound system. The excitation of a linear harmonic oscillator by a time dependent potential q2F(t) and the collinear collision of an atom and an anharmonic diatomic molecule are considered as examples. The results obtained for both these systems are in good agreement with the exact quantum mechanical transition probabilities even for low order transitions.
Starting from the study of one-dimensional potentials in quantum mechanics having a small distance behavior described by a harmonic oscillator, we extend this way of analysis to models where such a behavior is not generally expected. In order to obtain significant results we approach the problem by a renormalization group method that can give a fixed point Hamiltonian that has the shape of a harmonic oscillator. In this way, good approximations are obtained for the ground state both for the eigenfunction and the eigenvalue for problems like the quartic oscillator, the one-dimensional Coulomb potential having a not normalizable ground state solution and for the one-dimensional Kramers-Henneberger potential. We keep a coupling constant in the potential and take it running with a generic cut-off that goes to infinity. The solution of the Callan-Symanzik equation for the coupling constant generates the harmonic oscillator Hamiltonian describing the behavior of the model at very small distances (ultraviolet behavior). This approach, although algorithmic in its very nature, does not appear to have a simple extension to obtain excited state behavior. Rather, it appears as a straightforward non-perturbative method.
This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.
The 1D collision between an atom and a diatomic molecule is investigated using an algebraic approach. Closed expressions for the quantum mechanical excitation transition probabilities are obtained. A Morse potential for the diatomic molecule is used. The classical trajectories are obtained in the united atom limit by taking an average over a finite number of states using the matrix density formalism, which allows the problem to be reformulated in terms of a time-dependent Hamiltonian. The system is studied in the interaction picture, which permits us to carry out an algebraic description through the approximation of the interaction potential in terms of a linear combination of the generators of the su (2) algebra. Comparisons with exact quantum mechanical results for transition probabilities in different systems are presented.
Heisenberg's correspondence principle for non-relativistic matrix elements has been generalized: quantal transition amplitudes between strongly coupled states are expressed as Fourier components of integrals over classical trajectories. The new theory reduces to Heisenberg's correspondence principle in particular cases. The theory is applicable whenever the change in quantum number is small compared with the initial quantum number. In these situations it is more comprehensive than both quantal first order perturbation theory and the sudden approximation. Furthermore, compared with standard quantal methods, the evaluation of amplitudes is particularly simple. The theory is worked out for one- and three-dimensional separable systems and the generalization to a system of arbitrary dimension is indicated.
The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.