On exact analytical solutions for the few-particle Schrodinger equation. IV. The asymptotic form and normalisability of the wavefunction

Abstract

For pt.III see ibid., vol.16, nol.18, p.4237-53 (1983). Formal series solutions for the Schrodinger equations of few-particle systems contain infinitely many associated with the normalisability, i.e. square-integrability of the wavefunction. The energy is one of these parameters. A method for determining the parameters by examining the asymptotic behaviour of the series has been developed. No integration, matrix inversions or trial and error procedures are involved. The method is directly applicable to the 1S states of two-electron atoms. If the wavefunction is expressed as a multipole expansion in spherical polar coordinates the radial functions have asymptotic properties which give rise to relations between the parameters. Values assigned to the parameters not specified by these relations (one per multipole) ensure that the wavefunction is normalisable.