---
title: "de Broglie: a wavelength for matter, a speed for the packet"
description: "What did assigning a wavelength to an electron predict, and which wave speed describes its motion?"
canonical_url: "https://sandboxphysics.com/stories/quantum-evidence/de-broglie-matter-waves-phase-packets/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/de-broglie-matter-waves-phase-packets/"
last_modified: "2026-10-15"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# de Broglie: a wavelength for matter, a speed for the packet

Sandbox Physics · 2026-10-15

What did assigning a wavelength to an electron predict, and which wave speed describes its motion?

## Turn the light question around

Light had already been given particle-like relations for energy and momentum. De Broglie asked the reverse question: did matter, until then described as particles, also need a wave description? His 1929 lecture looks back on a proposal made before Schrödinger’s wave mechanics had matured. It was a theoretical idea in search of observable consequences, not a picture of an electron rippling through space.

The proposed wavelength set the scale of any interference, and that scale should change when the electron’s momentum changes. Testing this dependence is sharper than noticing bright and dark regions on a detector, since many devices produce patterned images for unrelated reasons.

Read the evidence: L. de Broglie (1929), The Wave Nature of the Electron — Nobel lecture, December 12

## Momentum fixes the wavelength

The relation below links the wavelength to the magnitude of the momentum through Planck’s constant. For a free particle with mass, the momentum must be computed consistently with its energy; the nonrelativistic formula works only when the kinetic energy is small compared with the rest energy. The wavelength is not the physical diameter of the electron.

A plane wave has a single momentum and extends without end. A localized packet needs a spread of momenta, so it has a central wavelength but no single sharp wavelength everywhere. The wave-packet Lab explores this by solving a wave equation it takes as given; it does not derive the matter-wave hypothesis independently.

$$
\lambda=\frac{h}{p}
$$

Read the evidence: L. de Broglie (1929), The Wave Nature of the Electron — Nobel lecture, December 12

## Two speeds answer two questions

A crest of the wave and the envelope of the packet need not move at the same speed. For a narrow packet of a free particle on the positive-energy relativistic branch, the derivative of energy with respect to momentum gives the group velocity below. Here the energy includes the rest energy, the reduced Planck constant links angular frequency to energy, and the mass is positive.

When the energy includes the rest energy, the phase velocity of a massive particle is faster than light. That carries no faster-than-light message, because a single-frequency crest is not a localized signal anyone can control. Subtracting the constant rest energy changes the phase convention and the phase velocity, but leaves this group-velocity derivative unchanged.

$$
\begin{aligned}v_{\mathrm g}&=\frac{d\omega}{dk}=\frac{dE}{dp}=\frac{pc^2}{E}\\ E&=\sqrt{p^2c^2+m^2c^4}\end{aligned}
$$

Read the evidence: L. de Broglie (1929), The Wave Nature of the Electron — Nobel lecture, December 12

## A standing-wave clue, not yet a full atomic theory

De Broglie linked the allowed orbits to the requirement that the wave’s phase match up around a closed path. This turned the discrete atomic conditions into a question about waves. On its own, it did not give the full hydrogen wavefunction, the transition strengths or the spectra of atoms with many electrons.

Wave mechanics later replaced the literal orbits with an eigenvalue and boundary-value problem, and the hydrogen Lab uses that later framework. Comparing the two descriptions is useful because a suggestive early argument and a complete calculation are different achievements.

Read the evidence: L. de Broglie (1929), The Wave Nature of the Electron — Nobel lecture, December 12 E. Schrödinger (1926), Quantisierung als Eigenwertproblem, Erste Mitteilung, Ann. Phys. 79, 361–376 — German scan

## The test needs controlled changes

In their 1927 author abstract, Davisson and Germer report electrons scattered from nickel and compare the observations with wave predictions. This connects the wavelength idea to experiment. Crystal structure, surface preparation and angular detection need their own account of the apparatus, which is left to the electron-diffraction articles.

To judge such evidence, fix the geometry and ask whether one momentum relation predicts the changes across several settings. Agreement at one conveniently chosen angle is weaker than a consistent trend. A schematic ripple animation gives neither the measured intensities at each angle nor their uncertainties.

Read the evidence: C. Davisson and L. H. Germer (1927), Diffraction of Electrons by a Crystal of Nickel, Phys. Rev. 30, 705 — author abstract

## A wave description still needs a rule for outcomes

The wavelength relation does not say how a spread-out amplitude turns into a pattern of detection events. Born’s interpretation of collisions answers that separate question. Likewise, uncertainty comes from how the state is prepared and which observables are measured; it does not mean that every instrument is simply too crude.

Read this article as a bridge from early atomic quantization to wave mechanics, and then to probability. The historical success lies in quantitative predictions that fit together. It does not require picturing the packet as a small lump of matter, and it does not decide how the wavefunction should be interpreted.

Read the evidence: M. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11 L. de Broglie (1929), The Wave Nature of the Electron — Nobel lecture, December 12

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [Watch a wave packet move and spread](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
- [Compare the later wave-mechanics picture of hydrogen](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/)

## Sources and reading notes

We read de Broglie’s 1929 Nobel lecture from its PDF, because the web page failed to load, and the author abstract of Davisson and Germer’s 1927 paper. The equations use modern notation for a free particle. The diffraction apparatus and its data belong to the separate electron-diffraction articles. We do not reconstruct any historical intensity curve.

1. [L. de Broglie (1929), The Wave Nature of the Electron — Nobel lecture, December 12](https://www.nobelprize.org/uploads/2016/04/broglie-lecture.pdf)
2. [C. Davisson and L. H. Germer (1927), Diffraction of Electrons by a Crystal of Nickel, Phys. Rev. 30, 705 — author abstract](https://journals.aps.org/pr/abstract/10.1103/PhysRev.30.705)
3. [M. Born (1954), The Statistical Interpretation of Quantum Mechanics — Nobel lecture, December 11](https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf)
4. [E. Schrödinger (1926), Quantisierung als Eigenwertproblem, Erste Mitteilung, Ann. Phys. 79, 361–376 — German scan](https://gilles.montambaux.com/files/histoire-physique/Schrodinger-1926-1.pdf)

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