The comparative role of potential structure in classical, semiclassical, and quantum mechanics

The corresponding effects of features in the potential on classical, semiclassical, and quantum mechanics are probed using the technique of functional sensitivity analysis. It is shown that the classical and quantum functional sensitivities are equivalent in the classical (small ft) and harmonic limits. Classical and quantum mechanics are known to react in qualitatively similar ways provided that features on the potential are smooth on the length scale of oscillations in the quantum wave function. By using functional sensitivity analysis, we are able to show in detail how the classical and quantum dynamics differ in the way that they sense the potential. Two examples are given, the first of which is the harmonic oscillator. This problem is well understood by other means but is useful to examine because it illustrates the detailed information about the interaction of the potential and the dynamics which can be provided by functional sensitivity analysis, simplifying the analysis of more complex systems. The second example is the collinear H + H ₂ reaction. In that case there are a number of detailed and striking differences between the ways that classical and quantum mechanics react to features on the potential. For features which are broad compared to oscillations in the wave function, the two react in qualitatively the same way. The sensitivities are oscillatory, however, and there are phasing differences between the classical and quantum sensitivity functions. This means that using classical mechanics plus experimental data in an inversion scheme intended to find the "true" potential will necessarily introduce sizeable errors.