Quasiclassical Study of the Quantum Mechanical Two-Coulomb-Centre Problem

Abstract

Abstract. A recurrent scheme for finding the quasiclassical solutions of the onedimensional equation got by the separation of variables from the Schrödinger equation in prolate spheroidal coordinates has been elaborated. By means of this scheme, WKB expansion quasiclassical two-Coulomb-centre wave functions at large distances between the fixed positive charges (nuclei) have been constructed for the entire space of the negative particle (electron). Our method provides simple uniform estimates of the eigenfunctions at arbitrary internuclear distances R, R 1 included. In contrast to the perturbation theory, the interaction need not be very small in the quasiclassical approximation. Its applicability domain is hence wider. That permits analyzing qualitative laws for the behavior and properties of quantum mechanical systems.

Splitting of Potential Curves in the Two-Coulomb-Centre Problem

“…The perturbation theory is not sufficient in this area of the under-barrier electron motion. Therefore, we propose to use the quasiangular WKB-function [6,7] …”

“…The slightly modified perturbation theory has been used to determine the local behaviour of the solutions of the Z 1 eZ 2 problem near a certain nucleus [6,7]. We propose to employ the quasiclassical approach (the WKB method) to construct the asymptotic expansions of the angular CSWF in the internuclear region.…”

Semiclassical Treatment of High-Lying Electronic States of H2+

“…On the contrary, semiclassical, WKB-like methods for electronically bound energies and separation constants of H 2 + have also led to good agreement with quantum results, − and we chose to take that approach. Existing semiclassical treatments of H 2 + are either carried out for the general two-center Coulomb case or are refined for accurate treatment of the lowest-lying states, or both.…”

“…The analyses can be complicated because of the presence of a potential barrier between the two nuclei. Some authors go around the classical turning points in the complex plane to derive quantization conditions that include reflection just above the barrier, − , whereas others utilize uniform approaches to derive quantization conditions that account for the coalescence of the classical turning points at the barrier top. ,,− Whereas these refinements are essential for the lowest-lying states of H 2 + , for higher states, only very small portions of the PECs involve energies near the height of the barrier, and a simple treatment of the system can supplant more refined methods with little loss of accuracy.…”

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

“…Интенсивное исследование этой задачи в последние сорок лет объясняется не только появлением мощных ЭВМ и успехами в развитии асимптотических методов решения обыкновенных дифференциальных уравнений, но и потребностями мезомолекулярной физики [2], [3] и теории ион-атомных столкновений [4]. Новые результаты были получены как для самой задачи о молекулярном ионе водорода H + 2 (см., например, работы [5]- [7] и имеющиеся в них ссылки), так и для задачи двух центров с сильно отличающимися зарядами [8]- [11]. Эта же задача для уравнения Дирака изучалась с помощью асимптотических методов в работах [12], [13].…”