The correspondence between classical nonlinear resonances and quantum mechanical Fermi resonances

The correspondence between classical nonlinear resonances and quantum mechanical Fermi resonances is discussed. The principle result is the recognition that the classical resonant behavior does not contribute to the diagonal matrix elements, but rather contributes to the off-diagonal matrix elements. As a direct consequence, it is not necessary for the classical dynamics corresponding to the quantum mechanical states involved in a Fermi resonance to exhibit resonant behavior. Instead, it is the classical dynamics associated with the matrix element which connects the quantum mechanical states involved in the Fermi resonance which must exhibit resonant behavior. These results are illustrated numerically using a very simple model of two kinetically coupled Morse oscillators.