Matter waves

Matter waves associate a wavelength with a particle's momentum. This does not mean a tiny ball wiggles along the drawn curve. The curve is a component of a complex probability amplitude: it carries phase, and its squared magnitude can predict where repeated measurements are likely to occur.

Think like a programmer

Store the wave state as complex samples, not as a list of particle positions. The renderer can show one real component, but the observable distribution comes from re * re + im * im after normalization. Keep momentum in kilogram metres per second and wavelength in metres until the display boundary; unit conversions belong in one visible function.

Model checklist

Inputs: Momentum p in kg·m/s, Planck's constant h in J·s, and a chosen spatial grid.
State: Complex amplitude ψ(x) at grid samples, plus a normalization total.
Rule: Compute wavelength h/p; update or construct phase; turn complex magnitude into a normalized probability density.
Output: Wavelength, phase pattern, and probabilities for positions or detector bins.
Check: Doubling positive momentum halves wavelength; probability samples sum to one after normalization.

De Broglie's relation is

\[\lambda=\frac{h}{p}\]

λ is wavelength in metres, h = 6.62607015 × 10⁻³⁴ J·s is Planck's constant, and p is momentum. For p = 6.62607015 × 10⁻²⁴ kg·m/s, the result is λ = 10⁻¹⁰ m, or 0.1 nm. That inverse relationship is a strong unit and direction test: larger positive momentum must make the wavelength shorter.

const PLANCK_CONSTANT = 6.626_070_15e-34; // J·s

function deBroglieWavelength(momentum: number) {
  if (momentum <= 0 || !Number.isFinite(momentum)) throw new RangeError("positive momentum required");
  return PLANCK_CONSTANT / momentum;
}

Matter-wave wavelength experiment

Change an electron-like particle's momentum. The line plots the real part of a plane-wave amplitude over three nanometres; it is not a path through space.

Momentum: 5 × 10⁻²⁴ kg·m/s

: Real part of complex amplitude (scaled for display)
: Zero-amplitude reference line

Momentum 5.0e-24 kg·m/s; de Broglie wavelength 0.133 nm.

The labelled momentum and de Broglie wavelength above are the complete result; the curve is only a visual aid for phase.

The plotted line is deliberately labelled as the real component of an ideal plane-wave amplitude. A global phase shift can change this particular line without changing its probability density. In code, preserve both components before reducing to an observable:

\[P(x)=\frac{|\psi(x)|^2}{\sum_j |\psi(x_j)|^2}=\frac{\operatorname{Re}(\psi)^2+\operatorname{Im}(\psi)^2}{\sum_j |\psi(x_j)|^2}\]

Use the denominator as a test oracle. If a finite-grid approximation produces a total far from one after the normalization step, inspect the boundary policy, grid spacing, or complex arithmetic before interpreting the picture.

Try this experiment

Prediction: Increasing momentum makes more phase cycles fit in the same three-nanometre view, while the plotted amplitude scale itself does not report a particle's location.

Set the momentum to 2 × 10⁻²⁴ kg·m/s, predict the wavelength, then change it to 8 × 10⁻²⁴ kg·m/s. Compare the displayed values and count the phase cycles. Explain why neither curve alone tells you where a single detection will occur.

Where this model breaks

The plane wave here extends forever, has exact momentum, and has no localized packet or potential. Real beams have finite spread, particles can be relativistic, and a detector needs an interaction model. A one-dimensional real-component plot also hides complex phase, spin, entanglement, decoherence, and the measurement process.

Summary

Matter-wave code begins with a typed inverse relationship: momentum determines wavelength. Keep the complex state long enough to model interference, normalize its squared magnitude before treating it as a probability, and never mistake an amplitude plot for a particle trajectory.

Glossary

de Broglie wavelength: wavelength associated with momentum.
Complex amplitude: phase-carrying wave value whose magnitude helps determine probability.
Probability density: normalized squared magnitude used to predict relative detection frequency.
Plane wave: ideal wave with one wavelength and unbounded spatial extent.

Self-check

What happens to de Broglie wavelength when momentum doubles?
Why is a real-component plot not a trajectory?
Which normalization check should hold for discrete probabilities?

Sources

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 Matter Waves, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Quantum Beginnings, 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 Matter Waves 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 Matter Waves into a test

Use normalized complex wave state to relate momentum, phase, interference, and probability density.

Name the inputs and units that the quantum beginnings 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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