Multiple-Time Perturbation Theory in Quantum Mechanics.

Emmanuel Fromager

Junhao Zhang

Shamik Banerjee, Jayanta K. Bhattacharjee

We construct a quantum mechanical perturbation theory which uses the multiple time scale technique. Working with the time translation operator, we use a variant on the method of Bender and Bettencourt. Our perturbation theory smoothly crosses over to the classical result as Planck's -->0. It is seen that this technique has a nonperturbative element built into it.

Jayanta K. Bhattacharjee, Deb Shankar Ray

Time-dependent perturbation theory in quantum mechanics is divergent at long times when the perturbation induces a resonance between two eigenstates of the unperturbed Hamiltonian. Divergences in perturbation theory are also common in quantum field theory and in critical phenomena. The renormalization group (RG) was designed to deal with these divergences. In the last two decades, this procedure has been extended to dynamical systems where the perturbation theory diverges in the long-time limit. In this article, we first review the connection between RG in the context of field theory and RG in the context of dynamical systems. We then show that the long-time divergence in the resonant situation in the time-dependent perturbation theory in quantum mechanics can be removed by using a RG-aided calculational scheme.

Luigi E. Picasso, L. Bracci, Emilio d’Emilio

Marco Frasca

The standard perturbation theory for time-dependent problems in quantum mechanics is reconsidered through renormalization-group methods. This approach justifies a posteriori the theory of quantum resonance given by the multiple-scale analysis of the perturbation series applied to the coupled equations of wave-function amplitudes. The resonance equations for the leading-order amplitude probabilties are then obtained up to second order and a general method for the algorithmic computation of higher-order terms of the perturbation series is given. The three-level model in a monochromatic wave with two resonant states is discussed in the light of the present results.