The higher orders of the theory of strong perturbations in quantum mechanics and the secularity problem

Summary

We solve the higher-order equations of the theory of the strong perturbations in quantum mechanics given inFrasca M.,Phys. Rev. A,45 (1992) 43, by assuming that, at the leading order, the wave function goes adiabatically. This is accomplished by deriving the unitary operator of adiabatic evolution for the leading order. In this way it is possible to show that at least one of the causes of the problem of phase-mixing, whose effect is the polynomial increase in time of the perturbation terms normally called secularities, arises from the shifts of the perturbation energy levels due to the unperturbed part of the Hamiltonian. An example is given for a two-level system that, anyway, shows a secularity at second order also in the standard theory of small perturbations. The theory is applied to the quantum analogue of a classical problem that can become chaotic, a particle under the effect of two waves of different amplitudes, frequencies and wave numbers.