Born-Oppenheimer approximation for resonances
Zusammenfassung der Projektergebnisse
Nonrelativistic quantum mechanics successfully describes spectra and dynamics of atoms and molecules. However, a more fundamental description takes into account also the coupling to the quantized radiation field, i.e. the coupling to photons. The standard model for non-relativistic particles coupled to the quantized radiation field is nowadays called the Pauli-Fierz model. In our project we studied a number of physically relevant problems related to this model. In the first part we showed that in the limit of small velocities and accelerations o the Pauli-Fierz model reduces back to the standard Schrödinger equation. We proved that the Pauli-Fierz model has solutions containing so called dressed states, that are particle states dragging with them a cloud of virtual photons, where the wave function of the dressed states satisfies the Schrödinger equation at leading order in a small parameter (the inverse mass of the particles). Moreover, we determined the second order corrections to the dynamics of dressed states, a mass renormalization due to the cloud of virtual photons and a velocity dependent interaction due to the finite speed of light. Technically this is an adiabatic limit: the dressed particles move slowly compared to the photons. But in contrast to the standard adiabatic setting there is no spectral gap in this problem. In the adiabatic approximation emission of photons, a small effect, is suppressed. To understand also emission of photons one thus needs to go beyond the adiabatic approximation and consider so called non-adiabatic transitions. We compute the leading order expression for the radiated power (energy per time) through emission of photons and find a formula which can be considered the quantum version of the Lamor formula of classical electrodynamics. The second part of the project was concerned with the dynamics of molecules. The basis for most qualitative and quantitative results in molecular dynamics is the Born-Oppenheimer approximation. The electrons in a molecule are eliminated from the dyo namical description and one is left with an effective Schr¨dinger equation for the nuclei only. The coupling to the radiation field is usually neglected, because radiative decay happens on a much larger time scale than molecular dynamics. As a mathematical problem the Born-Oppenheimer approximation ignoring the coupling to the quantized field is well understood by now. However, there have been no results at all justifying this basic approximation starting from the Pauli-Fierz Hamiltonian. Such a justification was obtained in this project. Moreover, we were able to prove also a formula for the rate at which transitions between different electronic levels through spontaneous emission of photons happen. It is a generalization of an analogous formula for the decay of resonances in atoms based on Fermi’s golden rule. The proof is rather delicate, since again this rates are very small and all other aspects of the dynamics need to be carefully controlled up to errors smaller than the terms of interest, i.e. the small emission rates. Moreover, the problem now contains three different time scales: the slow nuclei, the intermediate electrons and the fast photons.
Projektbezogene Publikationen (Auswahl)
- Effective dynamics for particles coupled to a quantized scalar field. Communications in Mathematical Physics 280 (2008), 751–805
L. Tenuta, S. Teufel
- Quasi-static limits in nonrelativistic quantum electrodynamics. Ann. Henri Poincaré 9 (2008), 553–593
L. Tenuta
