Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

==== Assume a boundary loop of length Lelem=2πreffL_{\rm elem}=2\pi r_{\rm eff}Lelem=2πreff with reff=r/2r_{\rm eff}=r/2reff=r/2. Quantization on the element gives ====

∮Celemp dl=p⋅(2πreff)=nh.\oint_{C_{\rm elem}} p\,dl = p\cdot (2\pi r_{\rm eff}) = n h.∮Celempdl=p⋅(2πreff)=nh.
Hence

p=nh2πreff=nℏreff.p = \frac{n h}{2\pi r_{\rm eff}} = \frac{n\hbar}{r_{\rm eff}}.p=2πreffnh=reffnℏ.
Physical angular momentum referenced to geometric radius rrr is

Lgeom  =  pr  =  nℏrreff  =  nℏrr/2=2nℏ.L_{\rm geom} \;=\; p r \;=\; \frac{n\hbar r}{r_{\rm eff}} \;=\; \frac{n\hbar r}{r/2} = 2 n\hbar.Lgeom=pr=reffnℏr=r/2nℏr=2nℏ.
If you instead count the elementary quantum nnn as half a unit when compared to a full geometric loop (i.e., n→s=12n\rightarrow s=\tfrac12n→s=21), you recover

Lgeom=sℏ,s=12,32,…L_{\rm geom} = s\hbar,\qquad s=\tfrac12,\tfrac32,\ldotsLgeom=sℏ,s=21,23,…
This small algebra shows how halving the physical radius shifts integer labels into half-integers for the full geometry.