The Leading Order of the Theory of Strong Perturbations in Quantum Mechanics

Abstract

We prove that, for a quantum system that undergoes a strong perturbation, the solution of the leading order equation of the strong field approximation (M.Frasca, Phys. Rev. A, {\bf 45}, 43 (1992)) can be derived by the adiabatic approximation. In fact, it is shown that greatest is the perturbation and more similar the quantum system is to an adiabatic one, the solution being written as a superposition of eigenstates of the time-dependent perturbation.A direct consequence of this result is that the solution of the Schrödinger equation in the interaction picture, in the same approximation for the perturbation, coincides with the one of the leading order of the strong field approximation. The limitation due to the requirement that the perturbation has to commute at different times is so overcome. Computational difficulties could arise to go to higher orders. Beside, the method is not useful for perturbations that are constant in time. In such a case a small time series is obtained, indicating that this approximation is just an application to quantum mechanics of the Kirkwood-Wigner expansion of statistical mechanics. The theory obtained in this way is applied to a time-dependent two-level spin model, already considered for the study of the Berry's phase, showing that a geometrical phase could arise if a part of the hamiltonian is considered as a strong perturbation. No adiabatic approximation is taken on the parameters of the hamiltonian, while their cyclicity is retained.