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Sandbox Physics

Q002 · Prepare / release / sample

Wave Packet & Uncertainty

Release a Gaussian wave packet, add a converging phase or combine two lobes. Watch the complex amplitude and both probability distributions. Estimate motion and spreading from independent position records.

Interactive modelWave Packet & Uncertainty
Model mean energy—\text{—}
Model uncertainty product—\text{—}
Recorded distributions and time series

SIMULATED INDEPENDENT PREPARATIONS

Build a distribution, one event at a time

Blue bars are recorded frequencies; the gray curve is a separate model density. Time-series dots show sample means with one standard error; orange dots show sample standard deviations. These are not confidence intervals or particle paths.

Current sample mean and spread

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Current measured uncertainty product

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Grid norm and boundary mass

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Velocity from position records

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Fourier propagation uses 2,048 points across a periodic interval of length 256. Edge mass is probability in the outer tenth of the grid. Any wraparound is a numerical boundary effect, not reflection from a real wall.

Model sources: MIT 8.04 · Correlated wave packets

All measured times and bases
Latest 12 events; CSV includes every event and preparation.

Physics tutorial

Wave packets are amplitudes

BackgroundA localized state contains a range of momenta. Their phases evolve at different rates.

Why it mattersA narrowing initial packet can broaden faster later; phase curvature can first make it contract.

Start with the essentials

Focus question
Can a free packet initially become narrower?
One-sentence intuition
Density width, phase curvature and mean motion are different controls.

Core mathematical model

Units

X=x/ℓ,P=pℓ/ℏ,τ=ℏt/(mℓ2)X=x/\ell,\quad P=p\ell/\hbar,\quad \tau=\hbar t/(m\ell^2)

Choose a length scale and particle mass. The displayed numbers use these dimensionless variables; the code uses unit mass and reduced Planck constant.

Free evolution

i∂τψ=−12∂X2ψ,ψ~(P,τ)=e−iP2τ/2ψ~(P,0)i\partial_\tau\psi=-\tfrac12\partial_X^2\psi,\qquad \widetilde\psi(P,\tau)=e^{-iP^2\tau/2}\widetilde\psi(P,0)

Every Fourier component evolves with its own phase. There is no potential, detector backaction during propagation, or physical reflecting wall.

Prepared lobe

ga(X)=exp⁡ ⁣[−(1−ic)(X−a)24σ2+ip0X]g_a(X)=\exp\!\left[-\frac{(1-ic)(X-a)^2}{4\sigma^2}+ip_0X\right]

The single state is a normalized central lobe. The double state adds lobes centered at opposite half-separations, with the chosen relative phase, then normalizes their coherent sum. Width labels describe each initial lobe, not the combined distribution.

Single-lobe analytic benchmark

⟨X⟩=p0τ,(ΔX)2=σ2+cτ+1+c24σ2τ2\langle X\rangle=p_0\tau,\qquad (\Delta X)^2=\sigma^2+c\tau+\frac{1+c^2}{4\sigma^2}\tau^2

Negative position–momentum covariance can initially focus the packet. This infinite-line expression is an independent benchmark for a single lobe; the simulator evolves the Fourier field for either preparation.

Uncertainty and covariance

(ΔP)2=1+c24σ2,CXP(0)=c2,ΔXΔP≥12(\Delta P)^2=\frac{1+c^2}{4\sigma^2},\quad C_{XP}(0)=\frac c2,\quad \Delta X\Delta P\geq\frac12

The product can exceed the lower bound even for a pure Gaussian because position and momentum are correlated. This is preparation uncertainty, not a statement about a simultaneous exact position-and-momentum readout.

Separate preparations

qˉ=1N∑jqj,sq2=∑j(qj−qˉ)2N−1\bar q=\frac{1}{N}\sum_j q_j,\qquad s_q^2=\frac{\sum_j(q_j-\bar q)^2}{N-1}

Each event starts from a new preparation. Position and momentum records are separate ensembles. One standard error estimates uncertainty of a sample mean; sample spread estimates preparation uncertainty.

Common difficulties

One dot or a cloud

Typical misconceptionA moving density is a known path of one particle.

Better mental modelThe cloud is a preparation distribution. Each recorded point comes from a new preparation, not continuous tracking.

Numerics versus inference

Typical misconceptionMore shots fix every error.

Better mental modelMore shots reduce sampling noise. They do not correct grid spacing, a periodic boundary, imperfect apparatus or a wrong physical model.

Run the experiment

  1. 01

    Release

    Play time and drag the initial width.

    What to observe: Narrow preparations spread quickly.
  2. 02

    Focus

    Choose Focus first and sample nine times.

    What to observe: The measured width first drops and then grows.
  3. 03

    Interfere

    Choose two lobes and change their relative phase.

    What to observe: The position and momentum distributions both encode coherence.
  4. 04

    Estimate

    Sample position and momentum at the same chosen time, then export.

    What to observe: Velocity uses only the recorded position means; preparation uncertainty uses separate ensembles.