The positive radial momentum operator

Abstract

The Laplacian in spherical coordinates contains the squared radial momentum operator $ p_r^2 = - r^{-1} \partial_r^2 r $ which is Hermitian and positive. However as has been pointed out by various authors the ``radial momentum operator" $ - i r^{-1} \partial_r r $ is not Hermitian. The positive square root operator of $ p_r^2 $ is found and also its inverse. We discuss the relation of these operators with Fourier transforms, the Hilbert transform and fractional integral operators.