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Journal of Physics B Atomic and Molecular Physics

Correspondence principle methods applied to a forced harmonic oscillator

A P Clark, A S Dickinson

Published Dec 1, 1971• 19 citations
Publisher

Abstract

The correspondence principle for strongly coupled states is used to obtain approximate transition probabilities for a forced harmonic oscillator. For particular forcing potentials, of interest in the vibrational excitation of molecules, the results are shown to be valid even for low order transitions.
DOI
10.1088/0022-3700/4/12/003

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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.
A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.
Transition probabilities for the forced, undamped quantum harmonic oscillator are obtained by expressing S-matrix elements entirely in terms of Heisenberg states and operators. The application of this method to this elementary-field theoretic model provides a simple example of the contraction technique of modern field theory whereby particles are converted into their corresponding currents. This problem was recently analyzed in the interaction picture from a pedagogical point of view and the treatment presented here is given in the same spirit.
When two atoms, held together firmly in a stable molecule are caused to vibrate, the vibrations may be treated approximately as motions of particles in a harmonic field. According to correspondence principle, at high quantum numbers, the oscillator should approach a macroscopic classical oscillator in behavior, e.g., a pendulum. The functions are particularly useful when discussing the screening effect of electrons in many electron atoms and the peaks of maximum probability corresponds to concentric shells strongly resembling the Bohr theory. The wave mechanical model on the other hand “spreads the electrons out” over all space, and so is in accord with Heisenberg’s principle.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.