--- title: "Airy Matter Waves Interactive Lab | Sandbox Physics" description: "A bending crest. No force. Give a wave a cubic phase mask. Its brightest ridge bends while its center of mass stays still. Reverse the mask, reverse the curve — and build the position pattern one detection at a time." canonical_url: "https://sandboxphysics.com/labs/airy-matter-wave-caustics/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/airy-matter-wave-caustics/" last_modified: "2026-10-18" --- FREE PARTICLES / CURVED INTERFERENCE # Airy Matter Waves Give a wave a cubic phase mask. Its brightest ridge bends while its center of mass stays still. Reverse the mask, reverse the curve — and build the position pattern one detection at a time. > Interactive simulation: open this page in a web browser to change the parameters and run the model. A finite, Gaussian-apodized one-dimensional wavepacket evolves under the free Schrödinger equation. This is a wavefunction diagnostic, not a particle path or a calibrated electron-beam instrument. The finite aperture eventually weakens and reshapes the Airy ridge. A SCULPTURE OF SPACE AND TIME BRIGHTEST POSITION $0$ GOLD LINE · CENTER OF MASS $0$ Phase hue · arbitrary origin 0 $2\pi$ LEFT → RIGHT: POSITION BACK → FRONT: TIME HEIGHT: AMPLITUDE This is a one-dimensional wave, shown across its complete history. Depth is time, not another spatial direction. Fixed amplitude gain; the visible position window is −40 to 40. $\tau=8$ 01 / WHAT REALLY MOVES ### A peak is not a center of mass. Cyan: brightest position. Gold: center of mass. The finite-aperture brightest lobe can switch; its curve need not remain a parabola. Drag to inspect a time. $\tau\ \longrightarrow\quad q=x/\ell$ 02 / SPACE AND TIME ### The interference draws the curve. Horizontal position −40 to 40; time 0 at top, 12 at bottom. Fixed square-root density color. Gold is the center; white is the selected time. 03 / POSITION ### A slice through the wave. Cyan: selected wave. Dashed coral: unshaped reference. Linear probability per bin, including overflow in the outer bins. $q=x/\ell$ 04 / MOMENTUM ### Same momentum weights. Different phases. Current wave and zero-mask reference have identical momentum probabilities. The cubic phase alone changes the spatial pattern. Vertical: momentum probability density. The displayed window is −4 to 4; moments use the full spectrum. $k=p\ell/\hbar$ FROM AMPLITUDES TO EVIDENCE ## One detection at a time. 0 / 8000 Every shot is a new independent preparation at the selected time. Position-grid Born sampling, all shots retained, fixed reproducible seed. The interval below covers shot noise only. Positions do not measure momentum or phase. Recorded fraction on the right $\text{—}$ 95% Wilson interval $\text{—}$ Model probability on the right $\text{—}$ Model momentum width $\text{—}$ ### Let the positions build the pattern. Gold: recorded fractions. Cyan: model probabilities. Position bins span −40 to 40; the first and last bins include every detection beyond the displayed window. Latest 20 records · export all trials | Trial | $q=x/\ell$ | Right half | | --- | --- | --- | The equation, the scale, the limits Position is measured in a freely chosen length unit; momentum and time follow from it. Positive position points right. No laboratory length, atomic mass or temperature is fitted. $$ q=x/\ell,\quad k=p\ell/\hbar,\quad \tau=\hbar t/(m\ell^2) $$ $$ i\partial_\tau\psi=-\tfrac12\partial_q^2\psi,\quad \widetilde\psi(k,\tau)=e^{-ik^2\tau/2}\widetilde\psi(k,0) $$ $$ \widetilde\psi(k,0)=C\exp(-ak^2+ibk^3/3) $$ $$ \langle q\rangle=-b/(4a),\quad\langle k\rangle=0 $$ The Gaussian spectral envelope makes the Airy packet normalizable. Its brightest ridge can curve while the center remains constant. A finite packet does not preserve its shape forever; the brightest lobe can change. The ideal fold curve is not used as the computed peak. 4096 Fourier points on a periodic interval of length 1024, spacing 0.25. Free evolution is a unitary spectral phase rotation; the lens is a position-phase multiplication. Normalization is performed only at preparation. The visible 80-unit window is a crop; the model, moments and position records use the full grid. Surface height is seven times the square root of position density. Density color saturates at 0.5; phase color uses an arbitrary global hue origin. Neither display choice changes the wave or the records. $\Delta N=0$ $P_{\mathrm{edge}}=0$ $P_{\mathrm{view}}=1$ Edge probability is measured in the outer 10% of the full numerical interval. Discretization and truncation are numerical errors, distinct from the displayed shot-noise interval. No interactions, environmental noise, detector blur or external forces between pulses. [Voloch-Bloch et al. · Generation of electron Airy beams · Nature (2013)](https://www.nature.com/articles/nature11840) Physics tutorial ## Airy Matter Waves **Background** Give a wave a cubic phase mask. Its brightest ridge bends while its center of mass stays still. Reverse the mask, reverse the curve — and build the position pattern one detection at a time. **Why it matters** Use a phase change to predict the future of a matter wave. Start with the essentials - **Focus question**: A bending crest. No force. - **One-sentence intuition**: The brightest interference crest is not the center of mass. ### Core mathematical model #### Free evolution $$ \widetilde\psi(k,\tau)=e^{-ik^2\tau/2}\widetilde\psi(k,0) $$ Momentum probabilities remain fixed between pulses; relative phases evolve. #### A finite Airy preparation $$ \widetilde\psi(k,0)=C e^{-ak^2+ibk^3/3} $$ A Gaussian spectral envelope gives a normalizable finite packet. Reversing the cubic phase mirrors it. ### Common difficulties #### A wave history is not a trajectory **Typical misconception** The particle travels across the raised surface. **Better mental model** Depth encodes time, height encodes probability amplitude, and each horizontal slice is a one-dimensional wavefunction. #### Respect the boundary **Typical misconception** A curved maximum proves a force on the particle. **Better mental model** A finite, Gaussian-apodized one-dimensional wavepacket evolves under the free Schrödinger equation. This is a wavefunction diagnostic, not a particle path or a calibrated electron-beam instrument. The finite aperture eventually weakens and reshapes the Airy ridge. ### Run the experiment 1. 01 #### Start with a prediction Compare all three presets at the indicated times. **What to observe:** The bright curve reverses with the phase mask; the center line stays straight. 2. 02 #### Locate the cause Scrub the timeline and compare position and momentum distributions. **What to observe:** Different cubic phases retain the same momentum weights, but produce different interference. 3. 03 #### Acquire and compare Record 400 detections, repeat, pin the run and export the ledger. **What to observe:** The histogram is computed from these positions; increasing shots reduces sampling noise. Physical conclusion A finite, Gaussian-apodized one-dimensional wavepacket evolves under the free Schrödinger equation. This is a wavefunction diagnostic, not a particle path or a calibrated electron-beam instrument. The finite aperture eventually weakens and reshapes the Airy ridge. 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)