The Helium Wave Equation

Abstract

This paper represents an attempt to solve the equation ${\ensuremath{\nabla}}^{2}\ensuremath{\psi}+(\frac{1}{z}){\ensuremath{\psi}}_{z}+(\frac{1}{4r})(E\ensuremath{-}V)\ensuremath{\psi}=0$, which is the Gronwall form of the wave equation for helium $S$ states. The equation ${\ensuremath{\nabla}}^{2}u+(\frac{1}{z}){u}_{z}=0$ is separable in polar coordinates ($r,\ensuremath{\beta},\ensuremath{\phi}$), and has solutions ${u}_{\mathrm{mk}}={r}^{2m\ensuremath{-}k}{sin}^{k}\ensuremath{\beta}{v}_{\mathrm{mk}}({sin}^{2}\ensuremath{\beta}){e}^{\mathrm{ik}\ensuremath{\phi}}={r}^{2m\ensuremath{-}k}{w}_{\mathrm{mk}}(\ensuremath{\beta},\ensuremath{\phi})$, where the ${v}_{\mathrm{mk}}$'s are jacobi polynomials, and the ${w}_{\mathrm{mk}}$'s form a complete orthogonal set of surface functions. The function $\ensuremath{\psi}$ is expanded as $\ensuremath{\psi}=\ensuremath{\Sigma}{\mathrm{mk}}^{}{\ensuremath{\psi}}_{\mathrm{mk}}({r}^{\frac{1}{2}}){w}_{\mathrm{mk}}$, resulting in an infinite system of ordinary linear differential equations for the ${\ensuremath{\psi}}_{\mathrm{mk}}$'s.