Probabilities for classically forbidden transitions using classical and classical path methods

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Abstract
Limits are established for the applicability of purely classical methods for calculating nonreactive, inelastic transition probabilities in collinear collisions of a structureless atom and a harmonic oscillator. These limits, obtained by comparison with previous exact quantum mechanical results, indicate that such methods are inappropriate not only for ’’classically forbidden’’ but for many ’’classically allowed’’ transitions (in spite of the fact that they are widely used to calculate probabilities for such processes). A classical path method in the context of infinite‐order time‐dependent perturbation theory is described which yields extremely accurate transition probabilities even for the most classically forbidden transitions in the collinear atom–harmonic oscillator system. The essential features of this method are: (1) the use of the expectation value of the total interaction potential in determining the atom–oscillator (central force) trajectory, and (2) the use of the arithmetic mean of the initial and final velocities of relative motion in the (elastic) central force trajectory. This choice of interaction potential allows the relative motion to be coupled to changes in the internal state of the oscillator. The present classical method is further applied to three‐dimensional atom‐breathing sphere collisions, and exact quantum mechanical calculations are also carried out. Comparison of the classical path and exact quantum results shows excellent agreement both in the specific inelastic cross section and in the individual partial‐wave contributions.