OBSERVATION CHAMBER / ACTIVE
Anderson Localization Observatory
Add seeded disorder to a finite quantum chain. Solve its eigenmodes and compare their density, participation, and concentration.
94 / STRUCTURE / QUANTUM / LOCALIZATION
Where does a stationary mode concentrate?
Canvas 2D is unavailable. All readings, the density table and spectrum remain available below. PNG is unavailable.
The density plot scales its height to the displayed peak; the density itself is normalized to sum 1. Touch the bottom spectrum to select the nearest energy. Focus the plot: ← / → mode, S solve, R reset.
EIGENMODE / NUMERICAL READOUT
Read one finite eigenstate
- Mode m
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- Energy E / J
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- IPR = Σ ρᵢ²
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- PR = 1 / IPR
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- Centroid (sites 1–32)
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- ‖Hψ − Eψ‖₂
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- Maximum eigenpair residual
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- max |QᵀQ − I|
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- Finite / acceptance
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- Jacobi sweeps
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A larger IPR means stronger concentration of this finite eigenstate; PR is its effective number of occupied sites. The clean chain has extended standing modes, not uniform density at every site.
Generated report
Site density and onsite potential
| i | W εᵢ | ρᵢ |
|---|
All eigenvalues and IPR
| m | E / J | IPR |
|---|
DETERMINISTIC SELF-TEST
Numerical acceptance: —
Clean N=3, 32, 64 eigenvalues and densities, independent 2×2 energies, seeded residuals and orthogonality, density normalization, IPR/PR bounds and reproducibility. The checks run separately from the current observation.
Acceptance details
OBSERVATION GUIDE
Compare a fixed realization
- Set W = 0 and press SOLVE. Select modes across the spectrum and compare their standing-wave densities.
- Keep the seed, raise W and solve again. Compare density, IPR and PR. Each solve uses the same εᵢ at that seed.
- Use NEW SEED to compare a different realization. REPORT records the current mode and complete spectrum; PNG saves the current chart. RESET clears the solution and outputs.
Finite size and the selected energy matter. An individual finite mode need not become monotonically more concentrated as W grows. Mode number follows energy order, not a tracked state identity.
MODEL NOTE / FORMULA
A real, finite, open chain
Hᵢᵢ = W εᵢ, Hᵢ,ᵢ₊₁ = Hᵢ₊₁,ᵢ = −J; J = 1; εᵢ ∈ [−½, ½); i = 1…32
Hψ = Eψ, Σ |ψᵢ|² = 1, IPR = Σ |ψᵢ|⁴, PR = 1/IPR, x̄ = Σ i |ψᵢ|²
W = 0: Eₘ = −2 cos[mπ/(N+1)], ψₘ(i) = √[2/(N+1)] sin[mπi/(N+1)]
There is no connection between sites 1 and N. Energies use J as the unit; site spacing is 1. Eigenvector sign is arbitrary. Density removes that sign choice. The numerical acceptance requires finite values, normalized density, IPR bounds, max residual ≤ 10⁻⁹ max(1, ‖H‖∞) and orthogonality error ≤ 10⁻⁹.
This is a static finite-chain observation model. It does not calculate transmission, a moving wavepacket, periodic Bloch gaps, infinite-system localization length, a mobility edge or a critical disorder strength. It does not measure a material.
P. W. Anderson — Absence of Diffusion in Certain Random Lattices (1958)
OBSERVATION POLICY
This finite open-chain tight-binding model runs in your browser. Settings, eigenmodes, reports, and images stay local. It does not measure a real material or visitors.
Route
This chamber belongs to Structures & Symmetries.
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