free space where the helicities are truly independent.

The measurable expectation (mean) values of the OAM, SAM, coordinate, momentum, and energy obtained from the diagonal operators (\[eqn:6\])–(\[eqn:8\]) can be written as $$\begin{aligned}
\label{eqn:10}
 {\bf S} = 
\left\langle {\tilde E^\sigma  } \right| \sigma {\bm \kappa }\left| {\tilde E^\sigma  } \right\rangle,~\\
\label{eqn:11}
{\bf L} = 
\left\langle {\tilde E^\sigma  } \right|{\bf \hat L}\left| {\tilde E^\sigma  } \right\rangle  - \left\langle {\tilde E^\sigma  } \right| \sigma {\bf A}_B \times {\bf k} \left| {\tilde E^\sigma  } \right\rangle,\\
\label{eqn:12}
{\bf R} = 
\left\langle {\tilde E^\sigma  } \right|i\partial _{\bf k} \left| {\tilde E^\sigma  } \right\rangle  - \left\langle {\tilde E^\sigma  } \right| \sigma {\bf A}_B  \left| {\tilde E^\sigma  } \right\rangle,\\
\label{eqn:13}
{\bf P}= \left\langle {\tilde E^\sigma  } \right|{\bf k}\left| {\tilde E^\sigma  } \right\rangle,~
W= \left\langle {\tilde E^\sigma  } \right|\omega\left| {\tilde E^\sigma  } \right\rangle.\end{aligned}$$ Here ${\bf A}_B  = {\bf e}_\phi k^{ - 1} \left( {1 - \cos \theta } \right)/\sin \theta$, convolution implies summation over $\sigma=\pm 1$ and integration in the ${\bf k}$-space, and we assume normalization $N=\left\langle {\tilde E^\sigma  } \right|\left. {\tilde E^\sigma  } \right\rangle=1$ (see Appendix for details). While the SAM is purely *intrinsic* (origin-independent), the OAM, in general, has both intrinsic and *extrinsic* contributions [@Extrinsic]: $$\label{eqn:14}
{\bf 