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Physical Review A

Exact solution of coupled Schrödinger equations in a stationaryn-state exponential model

Max Teubner

Published Jul 6, 2006• 1 citation
Publisher

Abstract

The $n$-state exponential model refers to coupled Schr\"odinger equations with potential ${V}_{ij}(x)={U}_{i}{\ensuremath{\delta}}_{ij}+{V}_{ij}{e}^{\ensuremath{-}\ensuremath{\alpha}x}$. Exact solutions in terms of Meijer's $G$ functions are obtained, provided the ${V}_{ij}$ factorize according to ${V}_{ij}=\ifmmode\pm\else\textpm\fi{}{V}_{i}{V}_{j}$. The nonadiabatic transition matrix $N$ is determined.

DOI
10.1103/physreva.74.012704

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The nnn-state exponential model refers to coupled Schr"odinger equations with potential Vij(x)=Ui\ensuremathδij+Vije\ensuremath−\ensuremathαx{V}_{ij}(x)={U}_{i}{\ensuremath{\delta}}_{ij}+{V}_{ij}{e}^{\ensuremath{-}\ensuremath{\alpha}x}Vij​(x)=Ui​\ensuremathδij​+Vij​e\ensuremath−\ensuremathαx. Exact solutions in terms of Meijer's GGG functions are obtained, provided the Vij{V}_{ij}Vij​ factorize according to Vij=\ifmmode±\else\textpm\fiViVj{V}_{ij}=\ifmmode\pm\else\textpm\fi{}{V}_{i}{V}_{j}Vij​=\ifmmode±\else\textpm\fiVi​Vj​. The nonadiabatic transition matrix NNN is determined.
A certain two-state exponential potential model is solved quantum mechanically exactly. Compact expressions for nonadibatic transition matrices are obtained. Interesting quantum mechanical threshold effects are found. Simple very accurate expressions are found from a semiclassical viewpoint for the nonadiabatic transition probabilities, indicating that the exponential model may present a third important basic model in addition to the Landau–Zener–Stueckelberg and the Rosen–Zener–Demkov models. Extension to general cases is also briefly discussed.
The nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.