Geodesic
The WKB method for the quantum mechanical two-Coulomb-center problem
Teoretičeskaâ i matematičeskaâ fizika, Tome 190 (2017) no. 3, pp. 403-418 Cet article a éte moissonné depuis la source Math-Net.Ru

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Using a modified perturbation theory, we obtain asymptotic expressions for the two-center quasiradial and quasiangular wave functions for large internuclear distances $R$. We show that in each order of $1/R$, corrections to the wave functions are expressed in terms of a finite number of Coulomb functions with a modified charge. We derive simple analytic expressions for the first, second, and third corrections. We develop a consistent scheme for obtaining WKB expansions for solutions of the quasiangular equation in the quantum mechanical two-Coulomb-center problem. In the framework of this scheme, we construct semiclassical two-center wave functions for large distances between fixed positively charged particles (nuclei) for the entire space of motion of a negatively charged particle (electron). The method ensures simple uniform estimates for eigenfunctions at arbitrary large internuclear distances $R$, including $R\gg1$. In contrast to perturbation theory, the semiclassical approximation is not related to the smallness of the interaction and hence has a wider applicability domain, which permits investigating qualitative laws for the behavior and properties of quantum mechanical systems.
Keywords: semiclassical approximation, WKB method, two Coulomb centers, asymptotic solution. 
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M. Hnatich; V. M. Khmara; V. Yu. Lazur; O. K. Reity. The WKB method for the quantum mechanical two-Coulomb-center problem. Teoretičeskaâ i matematičeskaâ fizika, Tome 190 (2017) no. 3, pp. 403-418. http://geodesic.mathdoc.fr/item/TMF_2017_190_3_a2/

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