MATTER WAVES
λ = h/p: an electron at 100 eV waves at 1.22637 Å — atom-sized, and the microscope is born.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Light turned out to carry momentum like a particle — Compton just showed it. De Broglie’s 1924 thesis bet the symmetry runs both ways: everything with momentum carries a wavelength, lambda equals h over p. A doctoral wager that won a Nobel prize.
The price
The momentum is priced relativistically and honestly: pc equals the square root of KE times KE plus two m c squared — exact at every speed, from thermal crawl to near-light. Divide hc by it and the wavelength falls out. Masses tabulated per particle, h and c exact.
The verdicts
An electron at a hundred electron-volts waves at 1.22637 angstroms — atom-sized. Crystals become diffraction gratings for electrons, and the electron microscope stops being a metaphor: the ruler at the bottom shows this wave sitting thousands of times finer than green light.
Four decades
The sweep runs the kinetic energy from ten electron-volts to a hundred thousand on a log clock: the packet tightens, the marker slides down the ruler past the atomic tick. And a thermal helium nucleus at room temperature — priced on the exact 2019 Boltzmann constant — still waves at seven-tenths of an angstrom. Matter waves are not an electron’s private trick.
Audit
Audited: the 100 eV electron locks at 1.22637 angstroms with the classical road off by 4.89 parts in a hundred thousand — quantified, not waved at — rising to 4.78 percent at 100 keV; the resolution chain reads 4484.80 times finer than green light; and the two momentum roads expose the site’s first conditioning lesson: algebraically identical formulas, but the invariant road cancels nine digits at helium scales while the model road holds machine precision.