SCIENCE · INVESTIGATE AND EXPLAIN

Compare matter wavelength with an observational scale

Calculate de Broglie wavelength from nonrelativistic kinetic energy and test when a wave description matters.

Learning outcome: Calculate de Broglie wavelength from nonrelativistic kinetic energy and test when a wave description matters.

Starting recap

Momentum p=mv and kinetic energy K=p²/(2 m) apply in this low-speed model.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

Compare a 100 eV electron and a 100 eV proton against a stipulated 100 pm lattice spacing.

Open the experiment

Predict

Do equal kinetic energies imply equal momenta and wavelengths?

Your prediction and reason

Experiment

Capture 100 and 400 eV for an electron, then 100 eV for a proton. Compare wavelength ratios and reported speeds as fractions of c.

Compare momentum and matter wavelength

Worked teaching model · synthetic values · Nonrelativistic free-particle p=√(2mK), λ=h/p. Masses are rounded NIST reference values. The 100 pm lattice is an original comparison, not an observed diffraction pattern.

Particle kinetic energy: 100 eV Particle: 0=electron, 1=proton: 0 choice

Reset keeps your prediction, captures and practice history.

Quantity	Calculated value	Unit
de Broglie wavelength	122.6426	pm
Comparison lattice spacing	100.0000	pm

Electron wavelength 122.64 pm.

Momentum 5.4027e-24 kg m/s; speed 5.9310e+6 m/s (0.0198c). Four times kinetic energy halves wavelength; this does not depict a particle following a wavy trajectory.

Model boundary: Nonrelativistic free-particle p=√(2mK), λ=h/p. Masses are rounded NIST reference values. The 100 pm lattice is an original comparison, not an observed diffraction pattern.

All investigations are virtual. Do not obtain radioactive sources, expose eyes to bright beams, or build high-voltage apparatus. Notebook captures preserve selected recipes, not physical measurements.

Electron wavelength 122.64 pm.

Momentum 5.4027e-24 kg m/s; speed 5.9310e+6 m/s (0.0198c). Four times kinetic energy halves wavelength; this does not depict a particle following a wavy trajectory.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Use a scale comparison rather than a slogan

A particle can produce interference evidence even though detections occur at localized places. The relation λ=h/p connects momentum to a wavelength; it does not say that the particle’s trajectory is a visible sine wave. Whether wave behavior matters to an observation depends on the apparatus scale and preparation.

Our comparison uses a hypothetical 100 pm spacing. A 100 eV electron has kinetic energy 1.602×10⁻¹⁷ J. With the stated rounded electron mass, p=√(2mK)≈5.403×10⁻²⁴kg m/s, giving λ≈122.64 pm. This is comparable to the spacing, so a purely point-trajectory model can miss diffraction structure. We have not calculated a crystal pattern or its intensities.

Change energy, then mass

At 400 eV the electron momentum doubles and wavelength falls to 61.32 pm. The relation is inverse square root, not inverse energy. At 100 eV a proton is much heavier, so it has greater momentum despite moving more slowly. Its wavelength is about 2.86 pm. Equal kinetic energy is not equal momentum, equal speed or equal wavelength.

The display reports speed/c as an assumption check. At the maximum electron energy of 1000 eV the nonrelativistic speed is about 0.0626c, making this a useful low-energy approximation rather than an exact relativistic calculation. At much higher energies, use the relativistic relation between energy and momentum; do not extrapolate p=√(2mK) indefinitely.

Transfer to a macroscopic object

For an original example, take a 0.020 kg object moving at 2 m/s. Momentum is 0.040 kg m/s and λ=h/p≈1.66×10⁻³² m. Compare that with a 1mm feature. The wavelength is extraordinarily small relative to the apparatus scale, explaining why an ordinary trajectory model works well for this comparison without asserting that quantum theory turns off.

Ratio check without constants

For two nonrelativistic particles of the same mass, λ₂/λ₁=√(K₁/K₂). Reducing energy to one ninth gives three times the wavelength. For equal kinetic energy but a mass four times larger, wavelength is halved.

What is not inferred

A wavelength is not an object’s size, a radius of orbit or a probability by itself. The experiment needs boundary conditions and a quantum state to predict an interference pattern. These numerical cases teach when assumptions should be questioned; no electron source or high-voltage construction is involved.

Source and authorship

Reference for the physical relationship. Source checked 30 October 2026. The explanations, numerical cases, questions and diagrams here are original; reference material is not copied. Exact h,c,e follow NIST SI definitions.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain why four times energy halves wavelength and why a matter wave is not a classical sinusoidal particle path.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate emitted-electron rate from stopping voltage Next: Conserve nucleon number and charge in nuclear change

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.