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.

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 origin0

2\pi

LEFT → RIGHT: POSITIONBACK → FRONT: TIMEHEIGHT: 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)

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 misconceptionThe particle travels across the raised surface.

Better mental modelDepth encodes time, height encodes probability amplitude, and each horizontal slice is a one-dimensional wavefunction.

Respect the boundary

Typical misconceptionA curved maximum proves a force on the particle.

Better mental modelA 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

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.
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.
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.