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

=== Goal: show how a geometric half-radius can produce half-integer phase conditions / spin-like behaviour using explicit boundary conditions and the action quantum h/2πh/2\pih/2π. ===

Start point (standard): for a wavefunction on a circle of radius RRR, angular coordinate φ∈[0,2π)\varphi\in[0,2\pi)φ∈[0,2π), a single-valued wavefunction ψ(φ)\psi(\varphi)ψ(φ) must satisfy

ψ(φ+2π)=ψ(φ),\psi(\varphi + 2\pi) = \psi(\varphi),ψ(φ+2π)=ψ(φ),
so plane-wave modes are ψm(φ)∝eimφ\psi_m(\varphi)\propto e^{im\varphi}ψm(φ)∝eimφ with integer mmm. Angular momentum eigenvalue Lz=mℏL_z = m\hbarLz=mℏ.

QAT picture: a spherical shell (2-D boundary) acts as the physical surface for the electron probability/charge. Consider motion (phase accumulation) around a great circle of that sphere. If the effective phase radius that matters to the phase accumulation is half the geometric radius — i.e. reff=r/2r_{\rm eff} = r/2reff=r/2 — then a full geometric rotation 2π2\pi2π corresponds to only half of the phase cycle on the effective circle. In other words, the phase coordinate ϕ\phiϕ that accumulates the quantum phase satisfies

ϕquantum=reffr φ=12 φ.\phi_{\rm quantum} = \frac{r_{\rm eff}}{r}\,\varphi = \frac{1}{2}\,\varphi .ϕquantum=rreffφ=21φ.
Thus a full geometric rotation φ↦φ+2π\varphi\mapsto \varphi+2\piφ↦φ+2π produces

ϕquantum↦ϕquantum+π.\phi_{\rm quantum}\mapsto \phi_{\rm quantum} + \pi.ϕquantum↦ϕquantum+π.
If the quantum boundary condition is antiperiodic on the effective variable (i.e. ψ\psiψ picks up a minus sign after 2π2\pi2π geometric rotation), then

ψ(φ+2π)=−ψ(φ).\psi(\varphi+2\pi) = -\psi(\varphi).ψ(φ+2π)=−ψ(φ).
Antiperiodicity corresponds to half-integer angular quantum numbers:

ψ(φ)∝eimeffφ,meff∈Z+12.\psi(\varphi)\propto e^{i m_{\rm eff}\varphi},\qquad m_{\rm eff}\in\mathbb{Z}+\tfrac12 .ψ(φ)∝eimeffφ,meff∈Z+21.
Then the angular momentum eigenvalue is Lz=meffℏ=(n+12)ℏL_z = m_{\rm eff}\hbar = (n+\tfrac12)\hbarLz=meffℏ=(n+21)ℏ and the smallest nonzero magnitude is 12ℏ\tfrac12\hbar21ℏ.

So the chain of geometric logic is:

• geometry: effective phase circumference is half the geometric circumference (reff=r/2r_{\rm eff}=r/2reff=r/2) ⇒ quantum phase completes only π for 2π geometric rotation;
• choose antiperiodic boundary condition (physical for spinors / half-spin objects) ⇒ half-integer modes;
• quantization: Lz=(n+12)ℏL_z=(n+\tfrac12)\hbarLz=(n+21)ℏ.

This is a short geometric way to see how “half” can arise from a half-radius / boundary-phase mapping. To make this rigorous you would: (i) derive the boundary action / boundary conditions for the surface wave and show they impose antiperiodicity (spinor-like), or (ii) show a physical reason (topological / gauge connection) why the effective phase coordinate is half the geometrical one (for example a ± sign from the gauge holonomy when you follow the field around the shell).

A compact formula tying the geometry to the action quantum:

∮effective looppφ dφ=2πmeffℏ.\oint_{\text{effective loop}} p_\varphi \,d\varphi = 2\pi m_{\rm eff}\hbar.∮effective looppφdφ=2πmeffℏ.
If the effective loop length is half the geometric circumference, the same geometric loop yields half-integer meffm_{\rm eff}meff.