Equation
LGV = I. Change the edges and you change LG, so the voltage and current pattern changes even when the injected current is unchanged.
Weighted resistor networknode brightness = voltage · edge width = |current|
Equivalent resistance—
Laplacian gap λ₂—
Total injected current1.000
Topology matters
A ring offers alternate paths; a chain does not. The equivalent resistance and spectral gap make part of that structural difference measurable.
What to test
Keep node count and conductance fixed. Switch only the topology. If the response changes, geometry—not the source amplitude—caused the change.
Ideal rectangular cavity — scalar TM-like modefield ∝ sin(mπx/a) sin(nπy/b)
Bars show the lowest relative mode frequencies. Tap a bar to select that (m,n) mode.
Selected mode(1,1)
Relative frequency—
Aspect ratio a/b1.50
Boundary conditions select modes
Changing a or b shifts the allowed eigenfrequencies because the field must still satisfy the cavity boundary conditions.
Degeneracy can appear or split
In a square cavity, modes such as (1,2) and (2,1) share the same idealized frequency. Stretching one dimension breaks that symmetry.
Model boundary
This is a scalar rectangular-cavity teaching model. Real electromagnetic cavities require the full vector Maxwell problem, materials, losses and three-dimensional geometry.
Tight-binding quantum networkcircle area = probability · sign ring = eigenmode phase
Tap an energy bar to inspect an eigenmode. Press “Localized dynamics” to return to time evolution.
Selected energy—
Participation ratio—
Time Jt/ℏ0.00
Hamiltonian
H = Σ εᵢ|i⟩⟨i| − JΣ⟨ij⟩(|i⟩⟨j|+|j⟩⟨i|). Connectivity determines which sites exchange amplitude.
Defects reshape modes
Increase the onsite defect and one or more eigenmodes can concentrate near that site. The graph did not move in physical space; the energy landscape changed.
Same structural lesson
Here the structure-dependent operator is H rather than a circuit Laplacian or Maxwell operator. Similar mathematics does not erase the physical distinctions.
What survives across all three?
Change structure → change operator → change modes → change observable behavior.
That is the common idea worth carrying between domains. The circuit uses a graph Laplacian, the cavity uses a boundary-value eigenproblem, and the quantum network uses a Hamiltonian. Each system keeps its own physics.