Adiabatic approximation, Gell-Mann and Low theorem and degeneracies: A pedagogical example

Christian Brouder (1) , Gabriel Stoltz (2) , Gianluca Panati (3)

1 IMPMC - Institut de minéralogie et de physique des milieux condensés
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]

We study a simple system described by a 2x2 Hamiltonian and the evolution of the quantum states under the influence of a perturbation. More precisely, when the initial Hamiltonian is not degenerate,we check analytically the validity of the adiabatic approximation and verify that, even if the evolution operator has no limit for adiabatic switchings, the Gell-Mann and Low formula allows to follow the evolution of eigenstates. In the degenerate case, for generic initial eigenstates, the adiabatic approximation (obtained by two different limiting procedures) is either useless or wrong, and the Gell-Mann and Low formula does not hold. We show how to select initial states in order to avoid such failures.

Mots clés: Gell-Mann Low; Many-body theory

Domaines: Electrons fortement corrélés [cond-mat.str-el]; Physique des Hautes Energies - Théorie [hep-th]

Christian Brouder, Gabriel Stoltz, Gianluca Panati. Adiabatic approximation, Gell-Mann and Low theorem and degeneracies: A pedagogical example. Physical Review A : Atomic, molecular, and optical physics [1990-2015], 2008, 78 (4), pp.042102. ⟨10.1103/PhysRevA.78.042102⟩. ⟨hal-00306471⟩