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.
Categories: quant-ph hep-th