Radial momentum operator ($p_r=-i\hslash ({\partial }_r+1/r)$)

For three-dimensional radial wavefunctions with measure $r^{2}dr$, this formal symmetric momentum expression obeys $[f(r),p_r]=i\hslash f^{\prime }(r)$ on appropriate test functions and has square $-{\hslash }^2({\partial }_r^2+2{\partial }_r/r)$. It is useful in separating kinetic energy as $p^2=p_r^2+L^2/r^2$. Formal symmetry and these identities do not establish a self-adjoint momentum observable on the half-line; domains and origin conditions must be specified when taking adjoints or defining products.