Q002 · Prepare, release and sample

Wave Packet & Uncertainty

Does a narrower wave packet end up spreading wider? Eventually, yes. A narrower packet contains a wider spread of momenta, so once released it spreads faster. Width and phase are set separately, though: give the packet a converging phase and it first narrows, then spreads.

Release a Gaussian wave packet, give it a converging phase or combine two lobes. Watch the complex amplitude together with the position and momentum distributions, and estimate how the packet moves and spreads from independent position records.

Model mean energy $\text{—}$

Model uncertainty product $\text{—}$

First, make a prediction

Does a free packet always get wider?

Drag the width handle. Play time forward, then try Focus first. Sample many fresh preparations to see what a detector would record.

Predict, compare, then collect evidence.

Measurement time — each sample uses a fresh preparation

PreparationInitial lobe width

\sigma

1

Carrier momentum

p_0

1

Position–momentum correlation

c

0

Observable to sample

Changing the time or the measurement basis keeps the records; changing the preparation clears them. Position and momentum are measured on separate preparations.

State details and sampling

Separation of two lobes

d

4

Relative phase

\varphi

0^{\circ}

Trials per time and basis

N

400

Sampling seed

s

1984

Position, momentum and time are dimensionless; the lesson gives the scale conversion. Sampling uses a reproducible seed and keeps at most 30,000 records.

The packet evolves under the free Schrödinger equation in one dimension, on a periodic Fourier grid. All scales are dimensionless. Position and momentum samples each come from fresh, separate preparations. The animation shows a probability distribution, not the trajectory of one particle.

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

\text{—}

Current measured uncertainty product

\text{—}

Grid norm and boundary mass

\text{—}

Velocity from position records

\text{—}

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

Start with the essentials

Focus question: Can a free wave packet get narrower at first?
One-sentence intuition: Width, phase curvature and mean motion are three separate features of the initial state.

Core mathematical model

Dimensionless units

X=x/\ell,\quad P=p\ell/\hbar,\quad \tau=\hbar t/(m\ell^2)

Pick a length scale and a particle mass. All displayed numbers use these dimensionless variables, and the code sets the mass and the reduced Planck constant to unity.

Free evolution

i\partial_\tau\psi=-\tfrac12\partial_X^2\psi,\qquad \widetilde\psi(P,\tau)=e^{-iP^2\tau/2}\widetilde\psi(P,0)

Each Fourier component picks up its own phase. During propagation there is no potential, no detector backaction and no physical reflecting wall.

One Gaussian lobe

g_a(X)=\exp\!\left[-\frac{(1-ic)(X-a)^2}{4\sigma^2}+ip_0X\right]

The single state is one normalized lobe at the center. The double state places a lobe half the separation to each side, adds them coherently with the chosen relative phase and normalizes the sum. The width setting refers to each initial lobe, not to the combined distribution.

Exact result for a single lobe

\langle X\rangle=p_0\tau,\qquad (\Delta X)^2=\sigma^2+c\tau+\frac{1+c^2}{4\sigma^2}\tau^2

A negative covariance between position and momentum can focus the packet at first. This formula holds on an infinite line and serves as an independent check for a single lobe; the simulation evolves the Fourier field for both preparations.

Uncertainty and covariance

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

Even a pure Gaussian can exceed the lower bound when its position and momentum are correlated. The relation describes the spread of the prepared state, not an exact simultaneous reading of position and momentum.

Separate preparations

\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, and position and momentum records come from separate ensembles. The standard error says how uncertain a sample mean is; the sample spread estimates the uncertainty of the prepared state.

Common difficulties

A particle or a cloud?

Typical misconceptionThe moving density is the known path of one particle.

Better mental modelThe cloud is the distribution of the prepared state. Each recorded point comes from a new preparation, not from tracking one particle continuously.

What more shots can fix

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

01
Release the packet
Play time and drag the initial width.
What to observe: A narrow packet spreads quickly.
02
Focus the packet
Choose Focus first and sample at nine times.
What to observe: The measured width first drops and then grows.
03
Make two lobes interfere
Choose two lobes and change their relative phase.
What to observe: Both the position and the momentum distributions carry the coherence between the lobes.
04
Estimate velocity and spread
Sample position and momentum at the same chosen time, then export.
What to observe: The velocity comes only from the recorded mean positions. The uncertainty of the prepared state comes from separate ensembles.

Wave Packet & Uncertainty

Build a distribution, one event at a time

Wave packets are amplitudes

Core mathematical model

Dimensionless units

Free evolution

One Gaussian lobe

Exact result for a single lobe

Uncertainty and covariance

Separate preparations

Common difficulties

A particle or a cloud?

What more shots can fix

Run the experiment

Release the packet

Focus the packet

Make two lobes interfere

Estimate velocity and spread

Related experiments and articles

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