Radial Potentials and the Hydrogen Atom

Abstract

If V is any radial function on ${\mathbb{R}}^{3}$, let $\hat{H} = -({\hslash }^{2}/(2m))\Delta + V$ be the corresponding Hamiltonian operator, acting on ${L}^{2}({\mathbb{R}}^{3}).$ We will look for solutions to the time-independent Schrödinger equation $\hat{H}\psi = E\psi$ of the form $\psi (\mathbf{x}) = p(\mathbf{x})f(\left \vert \mathbf{x}\right\vert ),$ where f is a smooth function on (0,∞) and p is a harmonic polynomial on ${\mathbb{R}}^{3}$ that is homogeneous of degree l.