Landau-Zener transitions for eigenvalue avoided crossings in the adiabatic and Born-Oppenheimer approximations. (English) Zbl 1079.81020

Rousse, Vidian

Asymptotic Anal. 37, No. 3-4, 293-328 (2004).

Summary: In the Born-Oppenheimer approximation context we study the propagation of Gaussian wave packets through the simplest type of eigenvalue avoided crossings of an electronic Hamiltonian \({\mathcal C}^4\) in the nuclcar position variable. It yields a two-parameter problem: the mass ratio \(\varepsilon^4\) between electrons and nuclei and the minimum gap \(\delta\) between the two eigenvalues. We prove that, up to first order, the Landau-Zener formula correctliy predicts the transition probability from a level to another when the wave packet propagates through the avoided crossing in the two different regimes: \(\delta\) being asymprotically either smaller or greater than \(\varepsilon\) when both go to 0.

MSC:

81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81V45	Atomic physics
81Q15	Perturbation theories for operators and differential equations in quantum theory
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics