  In this paper, we discuss some aspects of the energetics of a quantum
Brownian particle placed in a harmonic trap, also known as the dissipative
quantum oscillator. Based on the fluctuation-dissipation theorem, we analyze
two distinct notions of thermally-averaged energy that can be ascribed to the
oscillator. These energy functions, respectively dubbed hereafter as the mean
energy and the internal energy, are found to be unequal for arbitrary
system-bath coupling strength, when the bath spectral function has a finite
cutoff frequency, as in the case of a Drude bath. Remarkably, both the energy
functions satisfy the quantum counterpart of the energy equipartition theorem,
but with different probability distribution functions on the frequency domain
of the heat bath. Moreover, the Gibbs approach to thermodynamics provides us
with yet another thermally-averaged energy function. In the weak-coupling
limit, all the above-mentioned energy expressions reduce to $\epsilon =
\frac{\hbar \omega_0}{2} \coth \big(\frac{ \hbar \omega_0}{2 k_B T}\big)$,
which is the familiar result. We generalize our analysis to the case of the
three-dimensional dissipative magneto-oscillator, i.e., when a charged
dissipative oscillator is placed in a spatially-uniform magnetic field.
