An exact analytical solution in terms of the Meijer functions is obtained from the quantum Rosen-Zener-Demkov model for nonadiabatic transitions. The N matrix for the nonadiabatic transitions, the S matrices, and the probabilities for elastic and inelastic atomic collision channels are found.
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.
We derive an exact solution of an explicitly time-dependent multichannel model of quantum mechanical nonadiabatic transitions. Our model corresponds to the case of a single linear diabatic energy level interacting with a band of an arbitrary N states, for which the diabatic energies decay with time according to the Coulomb law. We show that the time-dependent Schrödinger equation for this system can be solved in terms of Meijer functions whose asymptotics at a large time can be compactly written in terms of elementary functions that depend on the roots of an N th order characteristic polynomial. Our model can be considered a generalization of the Demkov–Osherov model. In comparison to the latter, our model allows one to explore the role of curvature of the band levels and diabatic avoided crossings.
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 Vij factorize according to ${V}_{ij}=\ifmmode\pm\else\textpm\fi{}{V}_{i}{V}_{j}$. The nonadiabatic transition matrix N is determined.