Wave packets

A wave packet is a localized signal created by combining a range of wave numbers. It has an envelope—the broad region where amplitude is appreciable—and usually a faster carrier oscillation inside it. The packet is an array reconstruction from components, not a separate kind of particle-shaped wave.

Think like a programmer

Build the packet from an explicit spectral array: wave number, complex amplitude, and phase for each bin. Preserve that component list alongside reconstructed spatial samples. A renderer may draw one real component, but width, centre, normalization, and evolution checks must be calculated from the underlying state.

Model checklist

Inputs: Central wave number k₀, spectral width σk, component amplitudes and phases, domain, grid spacing, time, and dispersion relation.
State: Complex spectral coefficients, reconstructed spatial samples, packet centre, width, norm, and boundary configuration.
Rule: Apply each component's phase evolution then sum or transform the coefficients onto the position grid.
Output: Carrier, envelope, centre, width, norm, and any dispersion-driven reshaping.
Check: A narrower σk produces a wider initial envelope; translating phase shifts moves the packet without changing its norm; refinement and a larger domain limit wraparound error.

For a Gaussian distribution of wave-number components around k₀, one useful initial real component is

\[y(x,0)=Aexp!\left[-\frac{1}{2}(\sigma_k(x-x_0))^2\right]cos(k_0(x-x_0)),\qquad Delta xproptorac{1}{sigma_k}\]

The formula exposes a key data trade-off: a small spectral spread σk has a broad envelope in position; a large spread creates a more localized packet. Exact uncertainty constants depend on which width definition and Fourier convention the code uses, so record those choices instead of comparing bare “width” numbers from different solvers.

Gaussian wave-packet sampler

Change the spectral width σk. The blue line is one real field component at t = 0; the pale curves are its Gaussian amplitude envelope, not a second pair of waves.

Wave-number spread σk: 0.7 rad/m

: Real component of the packet
: Positive and negative Gaussian envelope

Wave-number spread σk = 0.7 rad/m; envelope width scale 1/σk ≈ 1.43 m. The packet is more localized in space.

The text result states the wave-number spread and reciprocal spatial-width scale. The plotted carrier is a view of one real component only.

At t = 0, the sampler uses the Gaussian envelope and a carrier cos(k₀(x − x₀)). A full evolution step first changes each complex component by its own phase factor and then reconstructs position samples. In a nondispersive relation, the packet can translate without changing its ideal shape. In a dispersive relation, component phases separate and the output can broaden, tilt, or develop side structure. Do not infer group velocity from the carrier crests; estimate the envelope centre from a documented amplitude or probability-density measure.

Try this experiment

Prediction: Reducing σk broadens the position-space envelope while leaving the chosen carrier wave number unchanged.

Set σk to 0.3 and 1.2, predict the ratio of their width scales, then inspect the envelope. Identify the spectral array field that controls localization and the separate field that controls carrier spacing. Finally name a norm or energy-like diagnostic that must be recorded during time evolution.

Where this model breaks

Finite grids wrap or truncate packets; a periodic discrete transform can make a departing packet reappear at the opposite boundary. Sampling can alias a carrier that exceeds the grid's resolvable wave number, and filters can silently alter normalization. Dispersion, loss, nonlinear response, and boundary reflections can reshape the packet, so an initial envelope is not permanent. A quantum wavefunction also needs a probability interpretation and a physical potential model before it can be called a particle prediction.

Summary

Wave packets are reproducible superpositions with an intentional spectral envelope. Keep components and reconstruction separate, use reciprocal width as a testable prediction, measure the envelope from data, and record boundaries and norm before interpreting an animation.

Glossary

Envelope: slowly varying amplitude boundary.
Carrier: rapid oscillation inside an envelope.
Bandwidth: range of frequency or wave-number components.
Carrier wave number: central spatial oscillation rate inside a packet.
Spectral coefficient: complex amplitude assigned to one wave-number component.
Aliasing: a discrete-grid error that misrepresents unresolved variation.

Self-check

Which data structure makes a packet reconstruction reproducible?
What happens to spatial width when wave-number bandwidth narrows?
Why can carrier motion differ from envelope motion?
Which grid behavior can make a packet appear to return from the other side?

Sources

OpenStax, University Physics Volume 1, Wave Packets.
A. P. French, Vibrations and Waves, packet construction and group velocity.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Wave Packets, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Waves II — Analysis and Transformations, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Wave Packets as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Wave Packets into a test

Construct localized waves from explicit component envelopes and relate bandwidth to spatial spread.

Name the inputs and units that the waves ii — analysis and transformations model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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