STUDY #23 · 2026 · IN OBSERVATION

Quantum Billiards

A model-driven visual study of quantum chaos in a stadium billiard.

MOVING IMAGE — THE ORBIT THAT REFUSES TO DIE classical ray → wave packet → scar chord · k 29.26 / 26.46 / 27.15 · γ 1 · copper-stm

WHAT IS THIS

A billiard is the simplest mechanics there is: a point bouncing inside a boundary. In a circular table the motion is perfectly ordered forever; give the table two straight walls — the Bunimovich stadium — and it becomes provably chaotic: a single ray fills the table and forgets its history.

The quantum version asks what standing waves this chaotic table can hold. This study solves the cavity's eigenmodes numerically (hundreds of them, cross-checked against exact solutions, spectral statistics and Weyl's law) and superposes them in real time on the GPU — stationary scarred states, spectral chords, and a Gaussian wave packet launched along the unstable diamond orbit, evolved exactly by phase rotation.

the diamond scar chord — three scarred eigenstates in phase, the ghost at its brightest k 29.26 / 26.46 / 27.15 · in phase · γ 1 · copper-stm

Motif stadium billiard / chaotic eigenmodes / quantum scars / level statistics

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation A single classical ray forgets its path within a few bounces, and the high eigenmodes are statistically Gaussian random waves — yet the spectrum still counts every classical orbit (Fourier peaks at the orbit lengths, including one we had not planned), and some eigenstates carry up to 2.9x the ergodic average along unstable orbits: Heller's scars, the ghosts chaos could not erase.

Reference E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984); L. A. Bunimovich, Commun. Math. Phys. 65, 295 (1979).

Tools Python / NumPy / SciPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME FAMILY, THE TABLE TURNS CHAOTIC

Chladni draws the nodal lines of order. The stadium draws the ghosts chaos left behind.

Study #03 — Chladni Study #23 — Quantum Billiards

The domain an integrable plate — modes in closed form a provably chaotic table — no closed form exists; the modes are found numerically

The nodal lines a lattice — sand settles into geometry a tangle — statistically indistinguishable from random waves

The protagonist the dark nodal line, where the sand rests the bright scar ridge — probability clinging to a dead orbit

What survives chaos — (order was never threatened) the ghost of an unstable orbit, ×1.8–2.9 over the ergodic mean

PARAMETERS EXPLORED

param meaning effect on the image

γ = a/r straight-wall length over cap radius — the geometry switch for chaos 0 is the integrable circle with its concentric nodal rings; any γ > 0 is chaotic by theorem, but low modes still remember order; at γ 1 the speckle is complete

n, k eigenstate index and wavenumber — the pitch of the drum higher modes refine the nodal tangle toward random waves; scars surface in the middle of the ladder

symmetry class parity under the two axes (±x, ±y) the skeleton of the picture — whether nodal lines ride the axes or avoid them

c_n superposition coefficients — which states sing, and how a single state is a standing crystal; an energy window shimmers; a wave packet runs the classical orbit, shatters and recurs

orbit the classical periodic orbit a scar is measured against diamond, bouncing-ball, whispering gallery — which ghost is summoned

palette STM copper / cavity steel / dark plate / interference / silver grounding in the experiment cultures that made these waves visible

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

One eigenproblem, run as a drum. The table is a membrane; every picture is an exact superposition of its standing waves.

-\nabla^2 \varphi_n = E_n \varphi_n \;\; \text{in } \Omega, \qquad \varphi_n = 0 \;\; \text{on } \partial\Omega

The quantum billiard: the Helmholtz eigenproblem of the cavity (ħ = 1, 2m = 1, E = k²). The circle solves in Bessel functions; the Bunimovich stadium (γ = a/r > 0) has no closed form — and is chaotic by theorem.

\psi(x,t) = \sum_n c_n\, \varphi_n(x)\, e^{-i E_n t}

All motion is superposition: time evolution is exact phase rotation — no integration, any moment reachable. A single eigenstate stands still; an energy window shimmers; a Gaussian packet runs the classical orbit and shatters.

E \;\approx\; \lambda + \frac{dx^2\, \lambda^2}{16}

The study's numerical find: correcting the five-point Laplacian's dispersion collapses the Weyl-law area error from 1.04% to 0.05% (perimeter 16.7% → 0.7%) — the eigenvalue ladder is trusted before any image is kept.

A visual interpretation of the model, not a claim of scientific precision. The stadium modes are numerical (the curved wall converges at first order); the GPU field matches the reference computation to max|Δψ| = 6.9e-7, and a single mode at t = 0 is bitwise identical.

SELECTED STILLS — 7

the diamond scar chord — the ghost at its brightest k 29.26 / 26.46 / 27.15 · in phase · γ 1 · copper-stm

a whispering-gallery mode hugging the right cap k 30.03 · single eigenstate · crop x 1.3 · copper-stm

the bouncing-ball family — the marginal vertical states, ×2.92 over the ergodic mean k 39.25 · single eigenstate · cavity

Berry random waves — a 38-mode chord and no ghost k ≈ 36 · 38 modes · golden-angle phases · darkfield

the integrable ancestor — a circle mode keeps its caustics circle (m 0, s 6) · k 18.07 · silver

one classical ray, 130+ bounces — ergodicity over a dim spectral chord ray from (0.31, 0.17), θ 0.64 · 38-mode chord · darkfield

the scar chord at centre crop — the ridge as copper relief k 29.26 / 26.46 / 27.15 · view 2.3 · copper-stm

PROCESS — PARAMETER SWEEPS

The scars were not placed; they were found. A closure search on the classical table rediscovered 44 periodic orbits, and every one of 961 computed eigenmodes was scored by its mean density inside a thin tube around each orbit. The sheet is that ledger's top three per orbit: the bouncing-ball family at ×2.92 the ergodic mean, the long axis at ×2.73, the diamond at ×1.80 — and the three states of the diamond family are the chord the hero video crystallises into.

the scar ledger — top-3 eigenstates per classical orbit 961 modes · 4 symmetry classes · k ≤ 40 · tube width 0.12 · V-bounce ×2.92 / long-axis ×2.73 / diamond ×1.80

SIGNATURE — THE SPECTRUM HEARS THE ORBITS

Chaos erases the path. Interference keeps its ghost.

Three times over, the dead orbits come back. In the eigenstates: along the unstable diamond orbit the probability density runs ×1.80 the ergodic average — Heller's scar, with the bouncing-ball family at ×2.92 and the long axis at ×2.73. In the dynamics: a Gaussian packet launched on the diamond shatters in a few bounces, yet re-peaks every classical period (|C| ≈ 0.27 at T = 0.169), its time average redrawing the orbit.

And in the spectrum: Fourier-transform the fluctuations of the level density and peaks stand at the lengths of the classical orbits — L = 4.005 for the bouncing ball, 8.015 for the long axis, and a strong line at 9.195 that was never planned: the closure search had already found a trapezoid orbit of length 9.176, off by 0.019. The spectrum counts orbits we did not tell it about.

the length spectrum — level-density fluctuations ring at the classical orbit lengths L 4.005 (V-bounce, ×44) · 8.015 (long axis, ×26) · 9.195 vs found trapezoid 9.176 (Δ 0.019) · k ≤ 40

COLOUR = STM COPPER, CAVITY STEEL, DARK PLATE

The stadium's waves have been seen in real laboratories twice over. Electrons corralled on a Cu(111) surface stand in exactly these ripples under a scanning-tunnelling microscope — the copper-relief palette renders probability as an STM topograph, bright crests of presence over dark nodal channels of absence. Microwave cavities machined into stadium shapes obey the same Helmholtz equation; the steel-and-heat palette answers them.

The dark plates quote the numerical tradition begun by McDonald–Kaufman and Heller, whose false-colour scars first showed the ghosts.

the bouncing-ball family in cavity steel — the microwave experiment answered k 39.25 · cavity palette (Stöckmann–Stein tradition)

All palettes are artistic approximations of those instruments and materials, not measurements.

REFERENCES

L. A. Bunimovich. "On the ergodic properties of nowhere dispersing billiards." Communications in Mathematical Physics, vol.65, 295-312 (1979).
E. J. Heller. "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits." Physical Review Letters, vol.53, 1515-1518 (1984).
S. W. McDonald, A. N. Kaufman. "Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories." Physical Review Letters, vol.42, 1189-1191 (1979).
O. Bohigas, M. J. Giannoni, C. Schmit. "Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws." Physical Review Letters, vol.52, 1-4 (1984).
M. V. Berry. "Regular and irregular semiclassical wavefunctions." Journal of Physics A: Mathematical and General, vol.10, 2083-2091 (1977).
M. F. Crommie, C. P. Lutz, D. M. Eigler. "Confinement of Electrons to Quantum Corrals on a Metal Surface." Science, vol.262, 218-220 (1993).

INTERACTIVE STUDY

A small window onto the model behind the finished works: 24 curated eigenmodes of the chaotic stadium cavity, superposed by exact phase rotation. Climb the energy ladder from the ground state to random waves — on the way, the diamond-scar family keeps a dead classical orbit glowing. It is a deliberately simplified instrument, capped in resolution with a few curated knobs and no export, separate from the full 168-mode engine used to author the finished works.

SIMPLIFIED INSTRUMENTSTADIUM γ 1 · 24 EIGENMODES · EXACT PHASE ROTATION

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.