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Recent Results on Non–Adiabatic Transitions in Quantum Mechanics

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Recent Results on Non–Adiabatic Transitions in Quantum Mechanics George A. Hagedorn and Alain Joye Abstract. We review mathematical results concerning exponentially small corrections to adiabatic approximations and Born–Oppenheimer approxima- tions. Introduction The goal of this paper is to review recent results on exponentially small non- adiabatic transitions in quantum mechanics. In Section 1, we provide background information about adiabatic approximations. In Section 2, we discuss the determi- nation of non–adiabatic scattering transition amplitudes. In Section 3, we describe the time development of exponentially small non–adiabatic transitions. In Section 4, we turn to exponentially accurate Born–Oppenheimer approximations. Then in Section 5, we discuss the determination of non–adiabatic corrections to Born– Oppenheimer approximations in a scattering situation. 1. Adiabatic Background The adiabatic approximation in quantum mechanics concerns the time–depen- dent Schrodinger equation when the Hamiltonian depends on time, but varies on a very long time scale. Mathematically, this situation corresponds to the singularly perturbed initial value problem (1.1) i ? ∂t??(t) = H(t)??(t), ??(0) = ?(0), where t ? R, ? is a small parameter, and ??(t) belongs to a separable Hilbert space H. In a simple situation, the adiabatic approximation relies on two basic assump- tions. The first is a regularity condition on the Hamiltonian: R: The Hamiltonian H(t) is a bounded self-adjoint operator on H that depends smoothly on t.

  • optimal adiabatic projectors

  • exponentially small

  • hamiltonian

  • small non–adiabatic

  • gap ?

  • scattering regime

  • recent results

  • small

  • transition amplitude

  • adiabatic transition


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RecentResultsonNon–AdiabaticTransitionsinQuantumMechanicsGeorgeA.HagedornandAlainJoyeAbstract.WereviewmathematicalresultsconcerningexponentiallysmallcorrectionstoadiabaticapproximationsandBorn–Oppenheimerapproxima-tions.IntroductionThegoalofthispaperistoreviewrecentresultsonexponentiallysmallnon-adiabatictransitionsinquantummechanics.InSection1,weprovidebackgroundinformationaboutadiabaticapproximations.InSection2,wediscussthedetermi-nationofnon–adiabaticscatteringtransitionamplitudes.InSection3,wedescribethetimedevelopmentofexponentiallysmallnon–adiabatictransitions.InSection4,weturntoexponentiallyaccurateBorn–Oppenheimerapproximations.TheninSection5,wediscussthedeterminationofnon–adiabaticcorrectionstoBorn–Oppenheimerapproximationsinascatteringsituation.1.AdiabaticBackgroundTheadiabaticapproximationinquantummechanicsconcernsthetime–depen-dentSchro¨dingerequationwhentheHamiltoniandependsontime,butvariesonaverylongtimescale.Mathematically,thissituationcorrespondstothesingularlyperturbedinitialvalueproblem(1.1)i²∂tψ²(t)=H(t)ψ²(t)²(0)=φ(0),wheretR,²isasmallparameter,andψ²(t)belongstoaseparableHilbertspace.HInasimplesituation,theadiabaticapproximationreliesontwobasicassump-tions.ThefirstisaregularityconditionontheHamiltonian:R:TheHamiltonianH(t)isaboundedself-adjointoperatoronHthatdependssmoothlyont.1991MathematicsSubjectClassification.Primary81Q05,81Q15;Secondary81Q55,81Q20.Keywordsandphrases.Molecularquantummechanics,adiabaticapproximations,Born–Oppenheimerapproximations.SupportedinpartbyNSFGrantDMS–0303586.1
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