A physically motivated derivation of the Laplacian in terms of the total angular momentum operator

Abstract

The straightforward transformation of the three-dimensional Laplacian and of the total angular momentum operator from Cartesian to spherical polar coordinates is tedious, unrewarding, and prone to errors. An alternative derivation, starting from the total angular momentum operator in Cartesian coordinates and using the generator of homogeneous scaling, readily yields the expression for the three-dimensional Laplacian in spherical polar coordinates in a manner that accounts very explicitly for the role of the total angular momentum operator. The two-dimensional case is presented as a preparatory exercise, and the N-dimensional case, which may be useful in a graduate course on the few-body problem, is shown to be as straightforward as the three-dimensional one.