Derive matter-wave scaling from momentum rather than photon formulas

A wavelength can describe matter

A particle can produce localized detection events while an ensemble shows interference patterns. The de Broglie relation associates wavelength with momentum: λ = h/p. This is not a claim that the particle is a little sinusoidal material object or that it follows a circular path whose radius equals the wavelength. The model links measurable momentum scale to an interference scale.

Derive the energy dependence

For a free nonrelativistic particle, K = p²/(2m). Solving gives p = √(2mK), and substitution gives λ = h/√(2mK). At fixed mass, wavelength scales with K to the power −1/2. At fixed kinetic energy, it scales with mass to the power −1/2. These relations differ from photon energy E = hc/λ, where rest mass is zero.

Make a factor comparison first

Increasing electron kinetic energy from 10 to 40 eV multiplies K by four, momentum by two and wavelength by one-half. The particle speed also doubles in the nonrelativistic model. This ratio reasoning can be checked before calculating any absolute value and can expose a mistaken direct inverse-energy dependence.

Compare two particle masses

At equal kinetic energy, a proton has about 1836.15 times the electron mass. Its momentum is therefore about √1836.15 ≈ 42.85 times larger, and its wavelength about 42.85 times shorter. Its speed is smaller by the same factor. Equal kinetic energies do not imply equal momenta, speeds or interference wavelengths.

Check the approximation numerically

The activity reports v/c. Within the selected 10–100 eV range the electron speed remains below about 0.02c, making this a useful low-speed teaching approximation. At much higher kinetic energies, the nonrelativistic energy–momentum relation would need replacement. A formula does not become universally valid because it gives finite numbers outside its intended domain.

Transfer and evidence

If an interference feature is modeled as inversely related to momentum, propose a controlled comparison of particle energy while keeping geometry fixed. State the predicted ratio rather than a hardware procedure. Then explain why an observed feature also requires an interaction/geometry model before it can be interpreted as a wavelength. The page does not request construction or operation of a charged-particle beam.

Sources and model boundary

Conceptual references checked 30 October 2026: OpenStax quantum theory, NIST 2022 CODATA constants. All lesson prose, numerical packets and diagrams are original; no textbook figures or exercises are copied. Synthetic parameter choices are not measured atomic spectra. The activity does not request radiation sources, vacuum equipment, particle beams or chemical manipulation. Saved calculations remain public practice, not independent mastery or physical laboratory evidence.