On exact analytical solutions for the few-particle Schrödinger equation. II. The ground state of helium

Abstract

Analytical methods for solving the Schrödinger equation directly can be applied to few-particle systems. This is illustrated by deriving a solution to the first-order perturbation equation for the ground state of helium. This solution is the in form of a partial wave expansion in spherical polar coordinates with Legendre polynomials as the angular functions. The radial functions include polynomials and exponential integral functions. Arbitrary parameters in the formal solution, associated with the square-integrability of the wavefunction, are identified. Their values are deter­mined by a least-squares method. The same arbitrary parameters occur in formal solutions of the higher-order perturbation equations. It is evident that a similar treatment can be applied to these equations, and to the eigenvalue problem.