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October WSPC Proceedings Trim Size: 75in x 5in hagjoy12final

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Niveau: Supérieur, Doctorat, Bac+8
October 19, 2010 0:4 WSPC - Proceedings Trim Size: 9.75in x 6.5in hagjoy12final 1 Non–Adiabatic Transitions in a Simple Born–Oppenheimer Scattering System George A. Hagedorn? Department of Mathematics and Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061–0123 USA Alain Joye Institut Fourier Unite Mixte de Recherche CNRS-UJF 5582 Universite Grenoble I BP 74 F–38402 Saint Martin d'Heres Cedex, France We study non–adiabatic scattering transitions in the Born–Oppenheimer limit for a molecular Schrodinger operator in which the nuclei have one degree of freedom and the electron Hamiltonian is a 2? 2 matrix. Keywords: Born–Oppenheimer Approximation, Non–adiabatic Transitions, Molecu- lar Quantum Mechanics 1. Introduction We describe non–adiabatic transitions in a simple Born–Oppenheimer scattering system. The detailed proofs are long and technical. They can be found in Ref. 4. These transitions are difficult to study because they are exponentially small and cannot be determined by perturbation theory. We study scattering theory for the equation i 2 ∂? ∂t = ? 4 2 ∂2? ∂x2 + h(x)? (1) in the Born–Oppenheimer limit ? 0. Here we assume h(x) is a 2 ? 2 matrix that depends parametrically on x and has an analytic continuation to a sufficiently wide strip about the real axis.

  • times ?k

  • leading order

  • wave packets

  • time–dependent born–oppenheimer

  • electronic potential energy

  • energy cut

  • any transition

  • transition probability

  • times another

  • constant times


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October 19, 20100:4
WSPC - Proceedings Trim Size: 9.75in x 6.5in
hagjoy12final
1
Non–Adiabatic Transitions in a Simple Born–Oppenheimer Scattering System
George A. Hagedorn Department of Mathematicsand Center for Statistical Mechanics, Mathematical Physics, and Theoretical Chemistry Virginia Polytechnic Institute and State University Blacksburg, Virginia24061–0123 USA
Alain Joye Institut Fourier Unite´MixtedeRechercheCNRS-UJF5582 Universite´GrenobleI BP 74 F38402SaintMartindHe`resCedex,France
We study non–adiabatic scattering transitions in the Born–Oppenheimer limit for a molecularSchro¨dingeroperatorinwhichthenucleihaveonedegreeoffreedomandthe electron Hamiltonian is a 2×2 matrix. Keywords: Born–Oppenheimer Approximation, Non–adiabatic Transitions, Molecu-lar Quantum Mechanics
1. Introduction We describe non–adiabatic transitions in a simple Born–Oppenheimer scattering system. The detailed proofs are long and technical. They can be found in Ref. 4. These transitions are difficult to study because they are exponentially small and cannot be determined by perturbation theory. We study scattering theory for the equation 4 2 ∂ψ ∂ ψ 2 i =+h(x)ψ(1) 2 ∂t2∂x in the Born–Oppenheimer limit0. Here we assumeh(x) is a 2×2 matrix that depends parametrically onxand has an analytic continuation to a sufficiently wide strip about the real axis. We also assumeh(x) approaches limitsh(±∞) suffi-ciently rapidly, as Rex→ ±∞, uniformly in the strip. We further assume that the
Supported in part by National Science Foundation Grant DMS–0907165.