QUANTUM TIME / A PARTICLE IN A BOX

Quantum Carpet & Revivals

Release a localized wave inside perfectly reflecting walls. Drag its starting point and explore a full space–time carpet: mirrored packets, split copies and an exact return. Freeze a slice and build its distribution from independent position records.

SPACE × TIME

STARTFIXED WALLS

THE WAVE AT THIS MOMENT · FIXED DENSITY SCALE

Preparing the spectrum…

FREEZE A MOMENT. BUILD THE EVIDENCE.

Where did the wave land?

Each ideal readout uses a fresh identical preparation. Collecting pauses time.

Recorded trials

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Return probability · model

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Mean position · records

\text{—}

Total probability · model

1

MODEL / RECORDS / PINNED

Horizontal coordinate $x/L$

Teal: model. Gold: recorded frequency density. Dashed: pinned model. Equal-width bins.

RETURN / OVER ONE CYCLE

Overlap compares the complete complex state with its preparation. A mirrored density can have very small overlap.

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Model, units and what the picture means

The initial Gaussian is multiplied by a sine envelope to vanish at both walls. Its first 64 sine coefficients evolve with the exact quadratic energy spectrum. Width denotes the Gaussian parameter before the wall envelope.

E_n=\frac{\pi^2\hbar^2n^2}{2mL^2},\quad T_{\mathrm{rev}}=\frac{4mL^2}{\pi\hbar}

All time labels use the revival period. Records sample the displayed 384-bin density, normalized for the ideal detector. No finite resolution, thermal ensemble or instrument response is modeled. Display brightness uses a fixed square-root gain.

Original research：Loinaz & Newman (1999)

Start with the essentials

Focus question: A wave scatters. Time brings it home.
One-sentence intuition: Quadratic energy spacings let dispersed modes rephase. A full revival restores the complete wave, not just its mean position.

Core mathematical model

The energy spectrum

E_n=\frac{\pi^2\hbar^2n^2}{2mL^2},\quad T_{\mathrm{rev}}=\frac{4mL^2}{\pi\hbar}

Time is displayed as a fraction of this exact revival period. The rotor rotational constant is an energy.

The evolving packet

\psi(x,t)=\sum_{n=1}^{64}c_n\sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}\,e^{-iE_nt/\hbar}

The prepared Gaussian includes a sine envelope at the hard walls. The displayed state is its normalized truncated projection.

What the records estimate

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

Return probability is a complex-state model diagnostic. Position counts alone do not measure or reconstruct it.

Common difficulties

What the image is

Typical misconceptionOne particle in a one-dimensional infinite well. A normalized 64-mode sine expansion of a wall-compatible Gaussian; no environment, interactions or finite-wall leakage. The carpet is a model probability map, not a particle trajectory.

Better mental modelThe vertical coordinate is time, not a second spatial direction. Bright stripes are regions of high probability; they do not identify paths taken by individual particles.

Run the experiment

01
Predict, then jump
Compare the start, a quarter period, a half period and one full period.
What to observe: Split copies, a mirror image and a return are different states.
02
Change one cause
Drag the starting position or add launch momentum while keeping the walls fixed.
What to observe: Compare the entire time trace, not a single attractive frame.
03
Acquire and audit
Collect 500 independent readouts. Pin the sample, change time, restore it and export the raw CSV.
What to observe: Counts fluctuate. The standard error describes sampling only; the teal reference comes from the model.