The~WKB method for the~quantum mechanical two-Coulomb-center problem
Teoretičeskaâ i matematičeskaâ fizika, Tome 190 (2017) no. 3, pp. 403-418
Voir la notice de l'article provenant de la source Math-Net.Ru
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.
@article{TMF_2017_190_3_a2,
author = {M. Hnatich and V. M. Khmara and V. Yu. Lazur and O. K. Reity},
title = {The~WKB method for the~quantum mechanical {two-Coulomb-center} problem},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {403--418},
publisher = {mathdoc},
volume = {190},
number = {3},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2017_190_3_a2/}
}
TY - JOUR AU - M. Hnatich AU - V. M. Khmara AU - V. Yu. Lazur AU - O. K. Reity TI - The~WKB method for the~quantum mechanical two-Coulomb-center problem JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2017 SP - 403 EP - 418 VL - 190 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2017_190_3_a2/ LA - ru ID - TMF_2017_190_3_a2 ER -
%0 Journal Article %A M. Hnatich %A V. M. Khmara %A V. Yu. Lazur %A O. K. Reity %T The~WKB method for the~quantum mechanical two-Coulomb-center problem %J Teoretičeskaâ i matematičeskaâ fizika %D 2017 %P 403-418 %V 190 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2017_190_3_a2/ %G ru %F TMF_2017_190_3_a2
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/