Time-Dependent Perturbation and Exact Results for the Periodically Driven Quantum Harmonic Oscillator

Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(omegat) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude