  In this paper, we discuss a general procedure by which nonlinear power
spectral densities (PSDs) of the harmonic oscillator can be calculated in both
the quantum and classical regimes. We begin with an introduction of the damped
and undamped classical harmonic oscillator, followed by an overview of the
quantum mechanical description of this system. A brief review of both the
classical and quantum autocorrelation functions (ACFs) and PSDs follow. We then
introduce a general method by which the kth-order PSD for the harmonic
oscillator can be calculated, where $k$ is any positive integer. This
formulation is verified by first reproducing the known results for the $k = 1$
case of the linear PSD. It is then extended to calculate the second-order PSD,
useful in the field of quantum measurement, corresponding to the $k = 2$ case
of the generalized method. In this process, damping is included into each of
the quantum linear and quadratic PSDs, producing realistic models for the PSDs
found in experiment. These quantum PSDs are shown to obey the correspondence
principle by matching with what was calculated for their classical counterparts
in the high temperature, high-Q limit. Finally, we demonstrate that our results
can be reproduced using the fluctuation-dissipation theorem, providing an
independent check of our resultant PSDs.
