Averaging with a time-dependent perturbation parameter

Time-Dependent Perturbation Theory

Our starting point is the set of eigenstates n of the unperturbed Hamiltonian 0 n H n E n = , notice we are not labeling with a zero, no 0 n E , because with a time-dependent Hamiltonian, energy will not be conserved, so it is pointless to look for energy corrections. What happens instead, provided the perturbation is not too large, is that the system makes transitions between the eigenstates n...

Averaging of Time–periodic Systems without a Small Parameter

In this article, we present a new approach to averaging in nonHamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approa...

The quantum 2D-harmonic oscillator in 1:1 resonance with time-dependent perturbation Averaging applied to slowly varying quantum systems

In this paper we will study the 2D-harmonic oscillator in 1:1 resonance. We add a general thirdand fourth order perturbation which is slowly time-dependent, and are interested in the resulting interaction between the unperturbed orbits (states). By treating this system quantummechanically we get a linear system (Schrödinger’s equation) on the infinite dimensional space . We show that this syste...

Notes 32 : Time - Dependent Perturbation Theory

Time dependent perturbation theory in Quantum Cosmology

We describe radiative processes in Quantum Cosmology, from the solutions of the Wheeler De Witt equation. By virtue of this constraint equation, the quantum propagation of gravity is modified by the matter interaction hamiltonian at the level of amplitudes. In this we generalize previous works where gravity was coupled only to expectation values of matter operators. By a “reduction formula” we ...