The Helium Wave Equation

This paper represents an attempt to solve the equation ${\nabla }^2\psi +\left(\frac{1}{z}\right){\psi }_z+\left(\frac{1}{4r}\right)(E-V)\mathrm{}\psi =0\mathrm{}$, which is the Gronwall form of the wave equation for helium $S$ states. The equation ${\nabla }^{2}u+\left(\frac{1}{z}\right)u_z=0\mathrm{}$ is separable in polar coordinates ($r,\beta ,\varphi$), and has solutions $u_{\mathrm{mk}}=r^{2m-k}{sin}^k\beta v_{\mathrm{mk}}\left({sin}^2\beta \right)e^{\mathrm{ik}\varphi }=r^{2m-k}{w}_{\mathrm{mk}}(\beta ,\varphi )$, 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 $\psi$ is expanded as $\psi =\Sigma {\mathrm{mk}}^{}{\psi }_{\mathrm{mk}}\left(r^{\frac{1}{2}}\right)w_{\mathrm{mk}}$, resulting in an infinite system of ordinary linear differential equations for the ${\psi }_{\mathrm{mk}}$'s.