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

==== Start with the wave-phase condition (Bohr–Sommerfeld style): ====

∮Cp⋅dl  =  nh(action quantization, h ⁣= ⁣Planck).\oint_{C} p\cdot dl \;=\; n h \qquad\text{(action quantization, }h\!=\! \text{Planck}) .∮Cp⋅dl=nh(action quantization, h=Planck).
For a circular path of radius RRR with constant tangential momentum p=mvp=m vp=mv,

∮p⋅dl  =  p⋅(2πR)  =  2πmvR  =  2πL\oint p\cdot dl \;=\; p \cdot (2\pi R) \;=\; 2\pi m v R \;=\; 2\pi L∮p⋅dl=p⋅(2πR)=2πmvR=2πL
because L=mvRL=m v RL=mvR is the orbital angular momentum. So the quantization condition gives

L  =  nh2π  =  nℏ.L \;=\; n\frac{h}{2\pi} \;=\; n\hbar .L=n2πh=nℏ.
Now insert QAT’s half-radius idea by saying the physical boundary layer that the phase accumulates over is not the full geometric circle of radius rrr but an effective boundary circle of radius

reff  =  r2.r_{\rm eff} \;=\; \frac{r}{2}.reff=2r.
If the microscopic phase circulation happens around that half-radius circle reffr_{\rm eff}reff, then the same argument applied to the elementary closed loop gives

Leff  =  mvreff  =  nℏ⇒mvr2=nℏ.L_{\rm eff} \;=\; m v r_{\rm eff} \;=\; n \hbar \quad\Rightarrow\quad m v\frac{r}{2}= n\hbar .Leff=mvreff=nℏ⇒mv2r=nℏ.
Solve for the physical angular momentum referenced to the geometric radius rrr:

Lgeom=mvr=2nℏ.L_{\rm geom}= m v r = 2 n \hbar .Lgeom=mvr=2nℏ.
Now read the quantization of the full geometric loop: the minimal nonzero n=1n=1n=1 for the elementary loop produces Lgeom=2ℏL_{\rm geom}=2\hbarLgeom=2ℏ. But crucially you can choose different identification: if the elementary quantum of phase is tied to a half circular loop (so that the wavefunction returns to itself only after two such elementary loops), then one elementary loop corresponds to n=12n=\tfrac12n=21 when counted against the full geometric circle. In that counting the allowed values of the full-circle angular momentum are

Lgeom=sℏ,s∈{12, 32, 52,…}L_{\rm geom} = s\hbar,\qquad s\in\{ \tfrac12,\  \tfrac32,\ \tfrac52,\ldots\}Lgeom=sℏ,s∈{21, 23, 25,…}
i.e. half-integers.

So the basic semiclassical mapping is: if the phase’s fundamental closed path is half the geometric circle (manifested by reff=r/2r_{\rm eff}=r/2reff=r/2), the quantum of circulation for the full circle can be half an integer — hence half-spin.

(Short boxed equation summary)

∮Celemp⋅dl=nh,Celem≡half-circ. of radius r/2⇒Lfull=sℏ, s∈Z+12.\oint_{C_{\rm elem}} p\cdot dl = n h,\quad C_{\rm elem}\equiv\text{half-circ. of radius }r/2
\quad\Rightarrow\quad L_{\rm full} = s\hbar,\ s\in \mathbb{Z}+\tfrac12.∮Celemp⋅dl=nh,Celem≡half-circ. of radius r/2⇒Lfull=sℏ, s∈Z+21.
Important: this is a semiclassical interpretation — it shows how halving the effective loop length can re-label the integer quantum nnn so the full-circle quantum numbers appear half-integer.