Theoretical and Mathematical Physics

, Volume 190, Issue 3, pp 345–358 | Cite as

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

  • M. HnatichEmail author
  • V. M. Khmara
  • V. Yu. Lazur
  • O. K. Reity


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.


semiclassical approximation WKB method two Coulomb centers asymptotic solution 


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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • M. Hnatich
    • 1
    Email author
  • V. M. Khmara
    • 2
  • V. Yu. Lazur
    • 3
  • O. K. Reity
    • 3
  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Institute of PhysicsPavol Jozef Šafárik UniversityKošiceSlovakia
  3. 3.Department of Theoretical PhysicsUzhhorod National UniversityUzhhorodUkraine

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