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Half-flux ground-state crossing on the ring

Physical scope

Exact level comparison in the ideal spinless ring model with periodic wavefunctions. It does not include spin, finite-width effects, disorder, or an experimental measurement.

Example

In the ideal spinless ring model, at dimensionless flux $α = q Φ / (2 π ℏ) = 1 / 2$ the ground energy is twofold degenerate at the Fourier labels $n = 0$ and $n = 1$ . At $α = 0$ the ground state is unique at $n = 0$ . The scale $κ = ℏ^{2} / (2 m R^{2})$ is in joules and $α$ is dimensionless.

Example references

Aharonov–Bohm ring spectrum and flux periodicity

Spectrum and gauge periodicity of a periodic flux operator

Facts & Assumptions

Given: the ideal ring Hamiltonian, $κ = ℏ^{2} / (2 m R^{2}) > 0$ , and its dimensionless flux $α$ .

[A1]

Choice, Countable Choice, and Dependent Choice are retained from the periodic spectral suppliers (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]

Its complete Fourier eigenbasis has energies $E_{n} = κ (n - α)^{2}$ for $n \in Z$ (Aharonov–Bohm ring spectrum and flux periodicity, Spectrum and gauge periodicity of a periodic flux operator).

Verification

1.1givenA1F1algebra

At $α = 1 / 2$ , $E_{n} = κ (n - 1 / 2)^{2}$ . The nearest integers to $1 / 2$ are exactly $0$ and $1$ , and both have squared distance $1 / 4$ ; every other integer has distance at least $3 / 2$ . Therefore $E_{0} = E_{1} = κ / 4$ is the ground energy and no other Fourier label attains it. The complete basis in [F1] makes the ground eigenspace exactly two-dimensional, conditional on the ideal ring assumptions.

2.1A1F1algebra∎

At $α = 0$ , $E_{n} = κ n^{2} \geq 0$ , with equality exactly at the integer $n = 0$ . The eigenbasis in [F1] then gives a one-dimensional ground eigenspace, spanned by $e_{0} (ϕ) = (2 π)^{- 1 / 2}$ , again conditional on the ideal ring assumptions.

Half-flux ground-state crossing on the ring

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources