The non-self-adjointness of the radial momentum operator in n dimensions

Abstract

The non self-adjointness of the radial momentum operator has been noted before by several authors, but the various proofs are incorrect. We give a rigorous proof that the $n$-dimensional radial momentum operator is not self- adjoint and has no self-adjoint extensions. The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on $L^2[(0,\infty ),dr]$ which is not self-adjoint and has no self-adjoint extensions.

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