A self-adjoint decomposition of radial momentum implies that Dirac's introduction of the operator is insightful

TL;DR

This paper presents a self-adjoint decomposition of the radial momentum operator, showing that Dirac's introduction of the operator is insightful and makes the operator physically meaningful and measurable.

Contribution

It introduces a novel decomposition of the radial momentum operator into self-adjoint parts, clarifying its physical interpretation and measurement.

Findings

Decomposition of _{r}P_{r} into self-adjoint parts

Reveals the operator's physical measurability

Supports Dirac's original operator insight

Abstract

With acceptance of the Dirac's observation that the \textit{canonical quantization entails using Cartesian coordinates, }we examine the\textit{\ }% operator $\mathbf{e}_{r}P_{r}$ rather than $P_{r}$ itself and demonstrate that there is a decomposition of $\mathbf{e}_{r}P_{r}$ into two self-adjoint but non-commutative parts, in which one is the total momentum and another is the transverse one. This study renders the operator $P_{r}$ indirectly measurable and physically meaningful.