--- title: "Quantum Billiards Interactive Lab | Sandbox Physics" description: "A ball bounces. A wave interferes. Send a quantum wavepacket across an arena. Bend the wall, add an island and watch reflection carve a new interference landscape. Collect position detections from the very same wavefunction." canonical_url: "https://sandboxphysics.com/labs/quantum-wave-billiards/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/quantum-wave-billiards/" last_modified: "2026-10-18" --- MATTER WAVES / BOUNDARY DESIGN # Quantum Billiards Send a quantum wavepacket across an arena. Bend the wall, add an island and watch reflection carve a new interference landscape. Collect position detections from the very same wavefunction. > Interactive simulation: open this page in a web browser to change the parameters and run the model. An isolated particle on a finite square lattice with hard boundaries. The surface shows probability amplitude, not a fluid or particle trajectory. Curved walls are stair-stepped; this is not a continuum-converged quantum-chaos calculation or an STM simulation. COHERENT WAVEFIELD EFFECTIVE OCCUPIED SITES $0$ MODEL · SPATIAL SPREAD Height: square root of site probability, fixed gain. Gold circle: the preparation region. The surface is a wavefunction diagnostic, not a moving material sheet. $\tau=32$ 01 / THE FULL STORY ### How far does the wave spread? Effective occupied sites. Cyan: selected arena. Dashed coral: empty rectangle, same launch. Drag to inspect. $\tau=Jt/\hbar$ 02 / PHASE ### Same brightness can hide a different state. Hue: complex phase. Brightness: amplitude, fixed scale. The hue origin is arbitrary; relative phase matters. Top is positive vertical coordinate. 0 $2\pi$ 03 / SPACE AND TIME ### Follow every reflection. Horizontal: lattice column, left to right. Vertical: time, 0 at top and 64 at bottom. Color shows the model position marginal, with fixed square-root scale. 04 / POSITION READOUTS ### One preparation. One position. Counts from the recorded positions only, divided by total shots. Empty until you measure. The gold region is fixed; positions alone do not measure quantum-state overlap. SAMPLE THE WAVEFUNCTION ## Make the pattern out of detections. 0 / 8000 Each detection is a fresh independent preparation at the selected time. Ideal lattice-position measurement; all shots are kept. Fixed simulation seed, reproducible on reset. Measured fraction in gold region $\text{—}$ 95% Wilson interval $\text{—}$ Model probability in gold region $\text{—}$ Effective occupied sites $\text{—}$ ### Do the counts follow the wave? Gold bars: recorded horizontal-position frequency. Cyan: model marginal. Linear probability scale. Columns run from 0 to 63. Latest 20 records · complete CSV export | Trial | $x/a$ | $y/a$ | Gold region | | --- | --- | --- | --- | Model, units, numerical checks & sources $$ H_0/J=4I-\sum_{\langle i,j\rangle}(|i\rangle\langle j|+|j\rangle\langle i|),\qquad\tau=Jt/\hbar $$ A finite square lattice, spacing a, uniform hopping J and no interactions. The surrounding and island sites are removed. The diagonal stays four even at a boundary: a discrete Dirichlet wall. This finite lattice model is the quantitative claim; the usual continuum interpretation would require a separate mesh-convergence study. $$ \psi_j(0)=C\exp\!\left[-\frac{(x_j/a+10)^2+(y_j/a+4)^2}{25}\right]e^{0.9i[(x_j/a+10)\cos\theta+(y_j/a+4)\sin\theta]} $$ The truncated Gaussian is normalized once, at preparation. Horizontal coordinate increases rightward, vertical upward. Angle is counterclockwise from the positive horizontal axis. The circle has radius 18; stadium semicircle centers are at horizontal coordinates minus and plus 9; the rectangle has half-width 27 and half-height 18. The island is centered at (4, 0). All lengths use lattice spacing. $$ F(t)=|\langle\psi(0)|\psi(t)\rangle|^2,\qquad N_{\mathrm{eff}}=\frac{(\sum_jp_j)^2}{\sum_jp_j^2} $$ The gold region is a disk of radius four around the initial center. Its position count is distinct from phase-sensitive state fidelity. Colors saturate at site probability 0.025; height is 24 times its square root in scene units. Spacetime color saturates at horizontal marginal probability 0.2. These display gains never change propagation or counts. Exponential action uses norm-limited substeps, without renormalization. No environment, loss, continuum extrapolation, chaos exponent or thermodynamic claim. Sampling uses a fixed pseudorandom seed; intervals describe shot uncertainty only. $0$ [Doncheski et al. (2003) · Wave packets in two-dimensional quantum billiards](https://arxiv.org/abs/quant-ph/0307070) [Wisniacki et al. (2001) · Sensitivity to perturbations in a quantum chaotic billiard](https://arxiv.org/abs/nlin/0111051) Physics tutorial ## Quantum Billiards **Background** Send a quantum wavepacket across an arena. Bend the wall, add an island and watch reflection carve a new interference landscape. Collect position detections from the very same wavefunction. **Why it matters** Follow one state through a complete causal experiment. Start with the essentials - **Focus question**: A ball bounces. A wave interferes. - **One-sentence intuition**: A hard boundary redirects amplitudes. Their phases determine the interference after reflection. ### Core mathematical model #### Evolve one state $$ |\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle $$ Every surface, diagnostic and sampled position is derived from this state. #### Compare the full state $$ F(t)=|\langle\psi(0)|\psi(t)\rangle|^2 $$ This model fidelity includes phase; it is not the fraction detected near the starting point. ### Common difficulties #### A raised wave is a diagnostic **Typical misconception** The particle rides along a material surface. **Better mental model** The particle is confined to a two-dimensional lattice; height encodes square-root probability with fixed display gain. #### Finite model **Typical misconception** These images prove quantum chaos or the arrow of time. **Better mental model** An isolated particle on a finite square lattice with hard boundaries. The surface shows probability amplitude, not a fluid or particle trajectory. Curved walls are stair-stepped; this is not a continuum-converged quantum-chaos calculation or an STM simulation. ### Run the experiment 1. 01 #### See the cause Select the three presets and scrub the full timeline. **What to observe:** Boundary shape and the island change the reflected interference pattern. 2. 02 #### Inspect phase Toggle phase color, then inspect the phase map below the arena. **What to observe:** Hue is a phase convention. Spatial differences matter; global hue does not. 3. 03 #### Acquire evidence Collect positions, pin the experiment and export all trials. **What to observe:** The measured frequency converges toward the model density, but does not reveal all phases. Physical conclusion An isolated particle on a finite square lattice with hard boundaries. The surface shows probability amplitude, not a fluid or particle trajectory. Curved walls are stair-stepped; this is not a continuum-converged quantum-chaos calculation or an STM simulation. Quantum & fields · Starter path ## Start from a single measurement Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference. [Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/) 1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/) 2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/) 3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/) 4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/) 5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/) 6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/) [View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)