Solution of the Fermi--Ulam Model in the Case of Periodic Perturbation

We discuss the evolution of the state and the average energy of the Fermi--Ulam model in the case of periodic perturbation. By a perturbation technique, the time-dependent Schrodinger equation is solved and it is found that the particle will continuously absorb or radiate energy if the frequency of the oscillating wall meets the resonance condition. Usually, these two states cannot exist together at a certain frequency. However, there is an exception if the frequency is at some special values. We find these values and reveal that the energy for transmission has the minimum equivalent unit, which is in the form of a harmonic oscillator.