Quantum chaos
Explore computed billiard eigenstates, finite spectral gaps and matched classical/quantum kicked-rotor snapshots.
About this tool
Choose eigenstates, level gaps or the kicked rotor, enter parameters and Compute. Each view exposes only its relevant fields. A cancellable worker performs the calculation; solver progress follows its iteration budget and can finish early. Resizing only redraws stored data. Valid settings may be saved locally; reload waits for Compute.
Billiards solve −Δψ=λψ with Dirichlet walls. The rectangle has width a and height1; its discrete eigenvalues/modes are analytical on the chosen five-point grid. The disk has radius.45 centred at(.5,.5). Stadium endcaps have radius.28, centred at x=.36/.64,y=.5, joined by a rectangle. Curved boundaries use a centre-point staircase mask. Inverse subspace iteration with conjugate-gradient solves, reorthogonalization and Rayleigh–Ritz rotation returns modes only when relative residuals meet1e−6. This is a discrete approximation, not an exact continuous-boundary spectrum. Degenerate mode orientations are not unique. Σ|ψᵢ|²=1; colors show cell probabilities relative to their maximum, not an absolute continuum density. Mode indices start at1.
Level gaps are adjacent computed eigenvalues divided by their sample mean; degeneracies remain. This is global normalization, not a full local unfolding. Blue histogram density integrates to1 over all observed gaps, without moving the tail into the last fixed bin. Reference laws exp(−s) and(πs/2)exp(−πs²/4) are drawn on their physical density scale. A small low-energy spectrum mixing symmetry sectors cannot establish Poisson/GOE statistics or the BGS conjecture; the curves are comparison laws, not a classification.
The rotor uses H=p²/2+Kcosθ·Σδ(t−n), with unit period/inertia and effective ℏ. Classical kick then drift: p′=p+Ksinθ, θ′=θ+p′ modulo2π. Quantum evolution before a kick uses U=exp(−iℏm²/2)exp(−iKcosθ/ℏ). Unitary DFTs connect the angle grid to signed momentum indices p=ℏm. Both models start with fixed momentum ℏm₀ and uniform angle distribution, and are displayed after the same number of kicks. Classical dots are an ensemble snapshot, not samples at mixed times.
The finite quantum grid has periodic Fourier aliasing. Compare larger grids and shorter windows; boundary weight is a measured warning of possible truncation effects, not a proof of localization. Green bars are quantum momentum masses; orange masses use the same bins of width ℏ. Classical mass outside the sampled momentum band is reported without renormalizing the remaining mass. Norm preservation is checked without correcting it artificially. Finite-time pictures alone do not prove chaos, scars or dynamical localization.
PNG exports the calculated plots with axes and actual parameters. Numerical failures discard the result rather than substituting a precomputed picture.