Radial Ladder Operators in 3D Coulomb Interaction

Abstract

We show that to obtain the ladder operators for the non-relativistic quantum 3D Coulomb problem it is enough to solve a simple linear system of first-order differential equations. We first assume the most general form for the operators compatible with problem and after, by applying the basic commutation relations they must comply with, we first deduce and then solve a differential equations system; in this way we find the usual form of the Coulomb radial ladder operators. The computations are explicitly done only for the creation operator since for the annihilation operator they are almost the same.