At a glance
A wavepacket is a sum of plane waves, and this plot is the recipe for the sum: how much of each wavenumber k to add. You shape the recipe with a centre k0 and a width σ, and read its values with a marker.
What you'll learn
- What the amplitude A(k) means, and how a packet is built from plane waves.
- How the centre k₀ and the width σ shape the A(k) curve.
- How to read A(k) with the marker, and why an amplitude is not a probability.
- How the width of the band in k is linked to the width of the packet in position.
A(k) against k: a bell centred on k₀, with width σ. The pink marker reads A at any k you choose.
What's on screen
- The curve: the amplitude A(k) for every k from −20 to 20, with the area under it shaded blue.
- Pink marker: a point on the curve at the k you choose, with its value A(k) printed beside it.
- Formula window (sidebar): the formula the curve is drawn from.
See itStart from the opening curve, k₀ = 0 and σ = 1, and park the marker at k = 2. Watch for: one bell centred on k = 0, with the pink marker on its right flank reading A(2.00) = 0.14.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Central Momentum k₀ | The centre of the bell: the wavenumber carrying the most weight. Read as a momentum, it is p₀ = ħk₀. | −5 to 5 opens at 0 | |
| Width σ | How wide the bell is in k. | 1 to 4 opens at 1 | |
| Show marker | Shows or hides the pink marker and its readout. The marker position slider appears only while it is on. | on / off opens on | |
| Marker position k | Where the marker sits. You can also click on the plot, or drag the marker, to move it. | −20 to 20 opens at 0 | |
| Formula window | The formula the curve is drawn from. It shows the symbols, not the numbers you have set. | display | |
| Reset | Puts k₀, σ and the marker back at their opening values, and restarts the mode you are in. | top bar |
Fixed: the k axis always runs from −20 to 20, and k carries no units in the sim. There is no time dependence, so nothing moves on its own.
A(k) is the recipe for a packet
A single plane wave eikx has one definite wavenumber and fills all of space. To build something that sits in one place, you add many plane waves together. A(k) says how much of each one goes into the sum:
The packet ψ(x) as a sum over plane waves, each weighted by A(k). The factor 1/2π is a convention.
- k is momentum in disguise. By de Broglie's relation p = ħk, so k₀ is the packet's central momentum, p₀ = ħk₀.
- A(k) is an amplitude, not a probability. The probability density for k goes as |A(k)|², which is narrower than the bell you see. The sim plots A only.
See itSlide the marker out along the opening curve, from k = 0 to k = 3. Watch for: the marker's reading falling from 1.00 to 0.61, 0.14, then 0.01.
What the centre k₀ does
The sim draws this Gaussian. Its two terms play different roles:
The sim's formula for A(k).
- The bell. The term −(k − k₀)²/2σ² puts the centre at k₀. Changing k₀ only slides the bell along k.
- The offset. The constant k₀²/2σ² makes the exponent zero at k = 0, so A(0) = 1 for every k₀ and σ. The peak is then A(k₀) = ek₀²/2σ²: 1.00 at k₀ = 0, but 90.0 at k₀ = 3 with σ = 1.
The sim's own plot at k₀ = 0 (above) and k₀ = 3 (below), with σ = 1. Only the vertical scale changes.
Key idea. The vertical axis rescales to fit 1.1 × the peak, so read the tick numbers, not the height on screen. The offset is this sim's convention. A constant factor does not change the shape, the widths, or any normalised probability.
See itKeep the marker at k = 0 and slide the centre from k₀ = 0 to k₀ = 3. Watch for: the marker at k = 0 still reading 1.00 while the vertical axis climbs to about 99.
What the width σ sets
- The band. A has fallen to e−1/2 ≈ 0.61 of its peak at k₀ ± σ, and to e−2 ≈ 0.14 at k₀ ± 2σ. About 95% of the area under the curve lies within k₀ ± 2σ.
- The spread. The probability density |A|² is narrower. Its standard deviation is Δk = σ/√2, which is 0.71 at σ = 1.
- The peak. At fixed k₀ a wider band has a lower peak. At k₀ = 3 the peak is 90.0, 3.08 and 1.32 for σ = 1, 2 and 4. At k₀ = 0 the peak is 1 for every σ.
- The limit σ → 0 is a spike at k₀: one plane wave with one definite momentum. It fills all of space and cannot be normalised, so no real particle is in it. The slider stops at σ = 1 as a display choice, not a law.
The three widths at k₀ = 3, each divided by its own peak.
See itHold k₀ = 3 and double the width from σ = 1 to σ = 2, with the marker at the peak. Watch for: the bell twice as wide, and the marker's reading at the peak dropping from 90.0 to 3.08.
The packet in space
Adding up the plane waves of this A(k) gives a Gaussian packet in x, a ripple of wavenumber k₀ under a smooth envelope:
The packet that A(k) builds.
Δk = · Δx =
k₀
σ
These sliders move the sim too.
- The standard deviation of |ψ(x)|² is Δx = 1/(σ√2). A wider band in k gives a narrower packet in x.
- The product ΔxΔk = ½ for every σ, so ΔxΔp = ħ/2. A Gaussian sits exactly on the lowest value the uncertainty relation allows.
- The sim draws only the k side. Δx is calculated from A(k), not drawn.
A(k)² and the packet |ψ(x)|² it builds, at k₀ = 3, each scaled to a peak of 1. Arrows mark Δk and Δx.
See itSet σ = 1, then σ = 4, with k₀ = 3. Watch for: the band on screen growing four times wider while the computed Δx shrinks four times, from 0.707 to 0.177.
Where it matters
- Confinement costs momentum. An electron held within Δx = 1 nm, about the size of a molecular orbital, needs a spread Δk ≥ 5 × 10⁸ m⁻¹. That is a velocity spread of at least 58 km/s.
- Short light pulses. The same relation links time and frequency. A 10 fs (FWHM) Gaussian pulse at 800 nm cannot be a single colour: it spans about 44 THz, roughly 95 nm of spectrum.
- Large objects. A 1 g bead held to 1 μm has a velocity spread of at least 5 × 10⁻²⁶ m/s. The relation still holds, but nothing could measure it.
Go deeper: why the Gaussian is special
- The Fourier transform of a Gaussian is a Gaussian. That is why both A(k) and the packet have the same bell shape.
- ΔxΔk = ½ is the smallest product any shape can have. Other shapes give a larger product.
- The sim shows one snapshot. How the packet changes in time depends on how each plane wave's frequency varies with k, which this sim does not include.
Summary
In one line. A(k) is the recipe for a wavepacket: how much of each plane wave to add. The width of the recipe in k and the width of the packet in x are locked together, with ΔxΔk = ½.
Key points
- k is momentum, p = ħk. A(k) is an amplitude, and the probability density goes as |A|².
- k₀ slides the bell. σ sets its width, with Δk = σ/√2.
- A(0) = 1 is the sim's convention, so the peak ek₀²/2σ² and the vertical scale change. This is bookkeeping.
- The packet in x has Δx = 1/(σ√2), so ΔxΔk = ½ whatever σ is.
Common mix-ups
- "A(k) is the probability of finding wavenumber k." It is the amplitude. The probability density goes as |A|².
- "Moving k₀ right gives the packet more amplitude." The larger peak comes from the A(0) = 1 convention. Shapes, widths and probabilities stay the same.
- "A narrow band in k makes a narrow packet." The reverse holds: Δx = 1/(σ√2).
Things to try on your own
- Set k₀ = 4. Read the peak for σ = 1 and σ = 2 with the marker, and compare with ek₀²/2σ².
- Put the marker at k₀ + σ, then k₀ + 2σ. Check that A is 0.61 and 0.14 of the peak.
- Use the section 06 sliders to find the σ that gives Δx = 0.25.