---
title: "Wave Packet & Uncertainty Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/wave-packet-uncertainty/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/wave-packet-uncertainty/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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](https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/pages/lecture-notes/) · [Correlated wave packets](https://arxiv.org/abs/quant-ph/0502097)

Physics tutorial

## Wave packets are amplitudes

**Background** A localized state contains a range of momenta. Their phases evolve at different rates.

**Why it matters** A narrower starting packet can spread faster later, and a curved phase can make it shrink first.

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 misconception** The moving density is the known path of one particle.

**Better mental model** The 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 misconception** More shots fix every error.

**Better mental model** More 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 the packet

   Play time and drag the initial width.

   **What to observe:** A narrow packet spreads quickly.

2. 02

   #### Focus the packet

   Choose Focus first and sample at nine times.

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

3. 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.

4. 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.

Physical conclusion

Sample position and momentum from separate preparations, and keep the spread of the results apart from sampling error.

## Related experiments and articles

Quantum evidence · Reading path

### Laws and the formation of quantum mechanics

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **de Broglie: a wavelength for matter, a speed for the packet**](https://sandboxphysics.com/stories/quantum-evidence/de-broglie-matter-waves-phase-packets/)
2. [Lab **Hydrogen Orbitals & Spectra** Compare the later wave-mechanics picture of hydrogen](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/)
3. [Article **Matrices and waves: two routes to the same predictions**](https://sandboxphysics.com/stories/quantum-evidence/matrix-wave-mechanics-observables/)

Starter path · Step 3 of 6

## Start from a single measurement

[Next step → **Quantum Harmonic Oscillator** Start from the ground state, then mix in a second level or displace the state. When does the probability cloud move, and what do sampled positions show? Model · 20 min](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)

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)