Controlled paraxial spin–orbital labels and nonparaxial corrections — Examples

By [F1], $f_{\sigma ,\mathrm{\ell }}$ is a unit vector and vanishes unless $\mathrm{\mid }\mathbf{k}\mathrm{\mid }<{\kappa }_{+}$. Since the integrand is nonnegative, $E_0(f_{\sigma ,\mathrm{\ell }})=\mathrm{\hslash }\int \mathrm{\mid }\mathbf{k}\mathrm{\mid }\mathrm{\mid }{f}_{\sigma ,\mathrm{\ell }}{\mathrm{\mid }}^{2}d\nu \le \mathrm{\hslash }{\kappa }_{+}\int \mathrm{\mid }{f}_{\sigma ,\mathrm{\ell }}{\mathrm{\mid }}^{2}d\nu =\mathrm{\hslash }{\kappa }_{+}<\mathrm{\infty }$. The radial multiplier is the phase generator in [F3]; with $\mathrm{\mid }\mathbf{k}\mathrm{\mid }$ in inverse-length units and $c=1$, this functional has energy units.

Fix $0<{\theta }_0<\pi$ and an integer $m\ge 2$, and set $C_{{\theta }_0,m}=\{\mathbf{k}\in M:1\mathrm{/}m\le \mathrm{\mid }\mathbf{k}\mathrm{\mid }\le m,\theta (\mathbf{k})={\theta }_0\}$. On this annulus, the polar formula in [F4] and the chart formula in [F5] give $\nu (C_{{\theta }_0,m})=\frac{1}{2}{\int }_{1\mathrm{/}m}^{m}rdr{\int }_0^{2\pi }d\phi {\int }_{\{{\theta }_0\}}\sin \theta d\theta =0,$ because the singleton $\{{\theta }_0\}$ has one-dimensional Lebesgue measure zero; [F6] justifies the iterated integral on this finite-measure annulus. Since $C_{{\theta }_0}={\bigcup }_{m=2}^{\mathrm{\infty }}{C}_{{\theta }_0,m}$, the fixed-angle surface is $\nu$-null. For any fixed $r_0>0$, the polar formula also gives $\nu (S_{r_0})=\frac{1}{2}{\int }_{\{r_0\}}rdr{\int }_{S^2}d\mathrm{\Omega }=0$, since the radial singleton has one-dimensional Lebesgue measure zero and the sphere measure is finite. Thus both sets in the Statement are null, so an $L^2(M,\nu ;{\mathbb{C}}^3)$ vector supported on either is zero almost everywhere and cannot be normalized. The construction uses genuine finite radial and angular widths rather than a distributional delta-cone mode.

The same supplier gives $J_z{f}_{\sigma ,\mathrm{\ell }}=(\mathrm{\ell }+\sigma )\mathrm{\hslash }{f}_{\sigma ,\mathrm{\ell }}$ and helicity $\sigma$. Substitution into [F2] yields the stated exact projected-spin expectation and, by the orbital identity there, $⟨L_z^{\mathrm{\prime }}⟩=(\mathrm{\ell }+\sigma )\mathrm{\hslash }-⟨S_z^{\mathrm{\prime }}⟩$. The normalization and finite radial support are as established in step 1.1.

The angular support has $0\le \theta \le {\theta }_2$. The cosine monotonicity and quadratic bound proved in the preceding proposition give $\mathrm{\mid }⟨S_z^{\mathrm{\prime }}⟩-\mathrm{\hslash }\sigma \mathrm{\mid }\le \mathrm{\hslash }(1-\cos {\theta }_2)\le \mathrm{\hslash }{\theta }_2^2\mathrm{/}2$. The packet is normalized, so this is a model expectation bound, not an empirical statement.