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The Journal of Chemical Physics

Transition probabilities for the driven harmonic oscillator

Francisco Marcelo Fernández

Published Nov 15, 1987• 4 citations
Publisher

Abstract

A simple and straightforward method is presented for calculating transition probabilities for the driven harmonic oscillator. The procedure does not require the explicit calculation of neither the time-evolution operator nor the Green’s function. As a result, a recurrence relation for the transition probabilities is obtained.
DOI
10.1063/1.453497

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For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.
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.