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American Journal of Physics

Exact and Impulse Scattering of Weakly Coupled Probes by an Harmonic Oscillator

H. A. Gersch

Published Dec 1, 1972• 0 citations
Publisher

Abstract

Comparison is made between the exact and impulse approximation expressions for scattering of weakly coupled probes by a one-dimensional harmonic oscillator. These results are utilized to estimate the applicability of the impulse approximation to several experimental situations. A first correction term to the impulse approximation is evaluated.

DOI
10.1119/1.1987067

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Comparison is made between the exact and impulse approximation expressions for scattering of weakly coupled probes by a one-dimensional harmonic oscillator. These results are utilized to estimate the applicability of the impulse approximation to several experimental situations. A first correction term to the impulse approximation is evaluated.
Using Green′'′s function and operator techniques we give a closed expression for the response of a non-relativistic system interacting through confining, harmonic forces. The expression for the incoherent part permits rapid evaluation of coefficients in a 1/q expansion. A comparison is made with standard approximation methods.
The coupling to the continuum is studied in a one-dimensional problem that describes the interaction of a weakly bound composite object with a wall in a semiclassical approach. A transformed harmonic oscillator basis is introduced to provide an appropriate discrete and finite basis for treating the continuum part of the spectrum. The convergence of the scattering magnitudes is investigated as the number of states in the basis is increased. The role of bound-to-continuum and continuum-to-continuum coupling is investigated.
The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
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.