The time correlation function for a harmonic quantum mechanical system can be related to the time correlation function for a corresponding classical system. Although straightforward to derive and well known in other contexts, this relationship has been unappreciated in the context of vibrational relaxation, where time correlation functions obtained from classical molecular dynamics have been used to predict relaxation rates for a quantum solute in a classical solvent. : This inconsistent treatment-quantum solute, classical solvent-predicts a relaxation rate which is slower than if the entire system, both solute and solvent, were treated classically. We demonstrate that if the classical time correlation functions are rescaled to account for the ratio of quantum to classical fluctuations, providing a quantum mechanical treatment for the solute and the solvent, the relaxation rates and the entire absorption spectrum are the same as for a purely classical treatment. Our conclusions are valid when the solute and solvent dan be described by a set of effective harmonic normal modes, and can also be valid when;anharmonicities are present.