Matter waves · boundary design

Quantum Billiards

How does the shape of a quantum billiard change the pattern a reflected wave makes? The walls decide which reflected pieces of the wave overlap, and their phases decide whether those pieces add up or cancel. Launched the same way, a packet in a circle, a stadium or a stadium with an island spreads into a different pattern of bright and dark patches at the same moment. A ball follows one bouncing path; the wave reflects off the whole wall at once.

Send a quantum wavepacket across an arena. Change the shape of the wall or add an island, and watch the reflections build a new interference pattern. Then measure the particle's position again and again; every detection samples that same wavefunction.

The model is a single isolated particle on a finite square lattice with hard walls. The raised surface shows the probability amplitude; it is not a fluid or the path of a particle. Curved walls are approximated by steps, so this is not a quantum-chaos calculation converged to the continuum, and it does not simulate an STM.

COHERENT WAVEFIELD

Effective occupied sites

0

Model · spatial spread

Height shows the square root of site probability, with fixed gain. The gold circle marks the preparation region. The surface is a plot of the wavefunction, not a moving sheet of material.

\tau=32

01 / THE FULL STORY

How far does the wave spread?

Vertical axis: effective number of occupied sites. Cyan: selected arena. Dashed coral: empty rectangle with the 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.

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

Build the pattern from single detections.

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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

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95% Wilson interval

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Model probability in gold region

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Effective occupied sites

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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.

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Doncheski et al. (2003) · Wave packets in two-dimensional quantum billiards

Wisniacki et al. (2001) · Sensitivity to perturbations in a quantum chaotic billiard

Start with the essentials

Focus question: How does the shape of a quantum billiard change the pattern a reflected wave makes?
One-sentence intuition: A hard wall sends the amplitude off in new directions, and the phases decide how the reflected waves interfere.

Core mathematical model

Evolve one state

|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle

Every surface, diagnostic plot and sampled position comes from this one state.

Compare the full state

F(t)=|\langle\psi(0)|\psi(t)\rangle|^2

This fidelity is computed from the model and includes phase. It is not the fraction of detections near the starting point.

Common difficulties

The raised surface is only a display

Typical misconceptionThe particle rides over a real, material surface.

Better mental modelThe particle moves only on a two-dimensional lattice. The height shows the square root of the probability, scaled by a fixed display gain.

A finite model

Typical misconceptionThese images prove quantum chaos or the arrow of time.

Better mental modelThe model is a single isolated particle on a finite square lattice with hard walls. The raised surface shows the probability amplitude; it is not a fluid or the path of a particle. Curved walls are approximated by steps, so this is not a quantum-chaos calculation converged to the continuum, and it does not simulate an STM.

Run the experiment

01
Compare the presets
Select the three presets and scrub the full timeline.
What to observe: Boundary shape and the island change the reflected interference pattern.
02
Inspect phase
Toggle phase color, then inspect the phase map below the arena.
What to observe: The colors follow a phase convention. Phase differences across space are physical; the overall hue is not.
03
Collect detections
Collect positions, pin the experiment and export all trials.
What to observe: As detections pile up, their frequencies approach the model density, but they cannot reveal all of the phase.