The Wigner method in quantum statistical mechanics, which allows the derivation of the quantum-mechanical behavior of the properties of an "almost classical" system, has been applied to the determination of series expansions with respect to
h for a general correlation function and the moments related to its spectrum. Explicit expressions of the terms up to the order
h4 are given for the zeroth, second, and fourth moment in the case of a many-body system with a Hamiltonian $\mathit{H}=\frac{{\mathit{P}}^{2}}{2m}+\ensuremath{\Phi}(\mathit{R})$ and for variables which are functions only either of
P or
R coordinates. From these expressions the corrections to the classical behavior can be calculated via the classical molecular-dynamics simulation technique.
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 ℏ) 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+H2 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.