Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Aharonov–Bohm ring spectrum and flux periodicity

Physical scope

Exact spectrum of an ideal spinless particle constrained to a zero-width circle with static enclosed magnetic flux and periodic wavefunctions. Finite-width rings, interactions, apparatus, and experimental identification are outside scope.

Statement

Adopt the static ring model of a spinless particle of mass $m > 0$ kilograms and charge $q \neq 0$ coulombs constrained to a circle of radius $R > 0$ metres. Use $ℏ > 0$ in joule-seconds, flux $Φ$ in webers, and energies in joules. The angular coordinate is $ϕ \in R / (2 π Z)$ ; the single-valued wavefunction belongs to the periodic domain $H_{p e r}^{2} (S^{1})$ ; the scalar potential is zero; and a static tangential vector potential $A_{ϕ} = Φ / (2 π R)$ has circulation $\oint A \cdot d ℓ = Φ$ , where $Φ$ is flux in webers. Under the SI minimal-coupling assignment, the Hamiltonian is $H_{Φ} = \frac{ℏ^{2}}{2 m R^{2}} {(- i \partial_{ϕ} - α)}^{2}, α = \frac{q Φ}{2 π ℏ},$ on that periodic domain. Its energies are $E_{n} (Φ) = \frac{ℏ^{2}}{2 m R^{2}} (n - α)^{2}, n \in Z,$ with a complete periodic Fourier eigenbasis. For every $j \in Z$ , $Φ \mapsto Φ + 2 π ℏ j / q$ gives a unitarily equivalent Hamiltonian; a noninteger $α$ cannot be removed by a single-valued periodic gauge phase. These are conditional deductions of the ideal ring model.

Statement references

Unbounded vector potential and effective Pauli model

Periodic Sobolev domain for a circle

Spectrum and gauge periodicity of a periodic flux operator

Facts & Assumptions

Given: the ideal ring constraint, $m > 0$ , $q \neq 0$ , $R > 0$ , $ℏ > 0$ , static flux $Φ$ , zero scalar potential, periodic wavefunctions, and the SI minimal-coupling postulate.

[A1]

The ring is an ideal zero-width constraint with no radial or transverse degrees of freedom. The model adopts a single-valued periodic wavefunction and the periodic $H^{2}$ realization of the tangential minimal-coupling expression; the static flux is represented by $A_{ϕ} = Φ / (2 π R)$ .

[A2]

The Choice, Countable Choice, and Dependent Choice axioms are the exact mathematical hypotheses of the periodic spectral calculation (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

The minimal-coupling covariant momentum is $- i ℏ \nabla - q A$ in SI units; $m$ is the mass parameter and $q$ the charge parameter (Unbounded vector potential and effective Pauli model).

[F2]

The angular Hilbert space and periodic $H^{2}$ domain are those of Periodic Sobolev domain for a circle.

[F3]

For $κ > 0$ and $α \in R$ , the operator $κ (- i \partial_{ϕ} - α)^{2}$ has complete eigenbasis $(2 π)^{- 1 / 2} e^{i n ϕ}$ , energies $κ (n - α)^{2}$ , integer-shift unitary equivalence, and no periodic gauge multiplier removing a noninteger $α$ (Spectrum and gauge periodicity of a periodic flux operator).

Proof

1.1givenA1A2F1algebra

On the circle, arclength is $s = R ϕ$ , so the tangential derivative is $\partial_{s} = R^{- 1} \partial_{ϕ}$ . The circulation of the constant tangential potential is $\int_{0}^{2 π} A_{ϕ} R d ϕ = Φ$ by [A1]. Therefore the covariant tangential momentum supplied by [F1] is $- i ℏ \partial_{s} - q A_{ϕ} = \frac{1}{R} (- i ℏ \partial_{ϕ} - \frac{q Φ}{2 π}) .$ Dividing its square by $2 m$ gives the Hamiltonian in the Statement and the dimensionless parameter $α = q Φ / (2 π ℏ)$ .

2.1A1A2F2F3step 1.1algebra∎

Put $κ = ℏ^{2} / (2 m R^{2}) > 0$ . The operator in the Statement is $κ (- i \partial_{ϕ} - α)^{2}$ , so [F2] and [F3] give its periodic domain, complete Fourier eigenbasis, energy list, and spectrum. If the flux is shifted by $2 π ℏ j / q$ , then $α$ shifts by $j$ and the unitary multiplication $U_{j} f = e^{i j ϕ} f$ intertwines the two Hamiltonians by [F3]. The same fact [F3] shows that a smooth single-valued periodic phase removing $α$ exists only for integral $α$ . Thus the spectrum and its gauge periodicity follow conditionally from the declared ideal ring model; no finite-width correction or measurement is asserted.

Remarks

The loop orientation is increasing $ϕ$ , and positive flux means the circulation $\oint A \cdot d ℓ = Φ$ in that orientation. The figure is a schematic of the excluded flux core and the constrained path; it is not a geometrical proof of the spectral statement.

Aharonov–Bohm ring spectrum and flux periodicity

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources