The WKB method for the quantum mechanical two-Coulomb-center problem


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 ≥ 1. 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.

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Correspondence to M. Hnatich.

Additional information

This research was supported by the Ministry of Education and Science of the Russian Federation (Agreement No. 02.a03.21.0008) and the Ministry of Education, Science, Research, and Sport of the Slovak Republic (VEGA Grant No. 1/0345/17).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 190, No. 3, pp. 403–418, March, 2017.

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Hnatich, M., Khmara, V.M., Lazur, V.Y. et al. The WKB method for the quantum mechanical two-Coulomb-center problem. Theor Math Phys 190, 345–358 (2017).

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  • semiclassical approximation
  • WKB method
  • two Coulomb centers
  • asymptotic solution