On Cyclic Harmonic Oscillators

It is proven that the energy of a quantum mechanical harmonic oscillator with a generically time-dependent but cyclic frequency, $ω_{0}(t_{0})= ω_{0}(0)$, 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.

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