  It is proven that the energy of a quantum mechanical harmonic oscillator with
a generically time-dependent but cyclic frequency, $\omega_{0}(t_{0})=
\omega_{0}(0)$, cannot decrease on the average if the system is originally in a
stationary state, after the system goes through a full cycle. The energy
exchange always takes place in the direction from the macroscopic system
(environment) to the quantum microscopic system.
