The one-Coulomb-centre problem in the spheroidal coordinate system: asymptotic solutions

Abstract

The one-Coulomb-centre problem is considered in the prolate spheroidal coordinate system. The asymptotic expansions for the separation constant and the Coulomb spheroidal quasiradial and quasiangular wavefunctions are derived when the distances between the foci of the spheroidal system R are large. The constructed wavefunctions are in excellent agreement with the exact wavefunctions in intra-atomic space, when the condition is fulfilled. The formulae obtained in the present paper can easily be generalized for the case of the two-Coulomb-centre problem.

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

“…In this section we give the basic formulae for the asymptotic behavior of the two-Coulomb-centre quasiradial and quasiangular wave functions in the vicinity of the left-hand nucleus [20] which will be employed for sewing with the quasiclassical solutions of the Z 1 eZ 2 problem obtained in the underthe-barrier region (see Section 4). Let us assume that when R tends to infinity, λ has the same order as R. Then in the zero-order approximation (i.e.…”

“…To find the solution at large but finite values of the parameter R, following [20] we shall use the perturbation theory. In equation (5), we shall consider the energy as a parameter with a certain given value and the separation constant λ as an eigenvalue of the corresponding operator.…”

“…Here the matrix elements of the operator 1/μ are diagonal. From (10) we may calculate any order of the corrections to the eigenvalue and eigenfunction (for details see [20]). …”

“…Here n = n 1 + n 2 + m + 1 and for p = 1, 2 all of the c (p) n 1,2 +k coefficients have been found in [20]. The expression for the normalization constant C is too cumbersome and not given here.…”

Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates

“…Recent attention to the solution of the hydrogen atom problem in spheroidal coordinates [22][23][24][25][26][27][28] revived interest in solutions of this type, which have in turn stimulated interest in the Coulomb Sturmians basis set, obtained in spheroidal coordinates. In [17,18], the Coulomb Sturmian amplitude functions are derived in two limiting cases, at small and large R. The results were obtained on representing the unknown functions in terms of Coulomb spherical (small R) and Coulomb paraboloidal (at large R) Sturmians.…”