Restricted quantum-mechanical three-body problems. IV - Asymptotically exact wave functions of a helium-like in ion all singular points of its Schroedinger equation

Details Restricted quantum-mechanical three-body problems. IV - Asymptotically exact wave functions of a helium-like in ion all singular points of its Schroedinger equation Restricted quantum-mechanical three-body problems. IV - Asymptotically exact wave functions of a helium-like in ion all singular points of its Schroedinger equation

Mar 1986

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986ap%26ss.120...61b&link_type=abstract

Astrophysics and Space Science (ISSN 0004-640X), vol. 120, no. 1, March 1986, p. 61-95.

Astronomy and Astrophysics

Astrophysics

2

Quantum Mechanics, Schroedinger Equation, Three Body Problem, Wave Functions, Asymptotic Methods, Atomic Energy Levels, Coulomb Potential, Eigenvalues, Helium Ions, Hilbert Space, Muons, Perturbation Theory

Scientific paper

On the basis of a perturbative procedure in which the eigenfunctions of a helium-like ion are expanded in the Hubert space built up from the eigenfunctions of an electron in two fixed Coulomb charges, all asymptotic eigenfunctions are constructed for the Schrodinger equation of helium-like ions. If the nuclear charge Z1 is not less than 2, then our asymptotic considerations clarify the singularities of the Schrodinger equation of a helium-like ion (at r1 = 0, ~, r2 = 0, ~, r12 = 0, cx), while in the case of Z1 = 1 (negative hydrogen ion) r2 = 0 will not be treated in this paper. The established order (in r2) of asymptotics at 0 or in an exceptional case the zeroth-order term of the functions (actually the coefficients of an expansion of the desired eigenfunctions) as one of the electron coordinates (r2, the distance of the two fixed Coulomb charges, a parameter of the set of the basic functions) enables us to classify the eigenfunctions of a helium-like ion. This classification resembles the classffication scheme for one-electron configurations. The asymptotics for r2 -~ ~ indicate bounded, pseudobounded (auto-ionizing) and free states. (Doubly ionized continuum states are not discussed here.) The use of the asymptotic solutions is indicated for the complete solution of the problem which may be either numerical integration or a variational procedure. Neutral muonic helium is included in the discussion

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