How long is a matter wave?

De Broglie wavelength λ = h/mv for an electron, proton, alpha-particle or ball, laid on a ruler of atomic spacings and X-ray wavelengths, with λ against V plotted.

Accelerate an electron until λ = 0.135 nm (≈83 V), then show a proton through the same voltage has a wavelength about 43 times shorter.

Thinking a heavier or faster particle has a longer wavelength, or that its charge sets λ.

• λ = h/p is smaller for a heavier particle (large m) or a faster one (large v).
• A ball of mass 0.12 kg at 20 m s⁻¹ has p = 2.40 kg m s⁻¹ and λ = 6.63 × 10⁻³⁴/2.40 = 2.76 × 10⁻³⁴ m. No instrument can detect a length that small, which is why everyday objects never show wave behaviour.
• Example 11.3(a): an electron (m = 9.11 × 10⁻³¹ kg) at 5.4 × 10⁶ m/s has p = 4.92 × 10⁻²⁴ kg m/s and λ = 6.63 × 10⁻³⁴/4.92 × 10⁻²⁴ = 0.135 nm. That is comparable with X-ray wavelengths.
• Example 11.3(b): a ball of mass 0.150 kg at 30.0 m/s has p = 4.50 kg m/s and λ = 1.47 × 10⁻³⁴ m, about 10⁻¹⁹ times the size of a proton: far beyond measurement.
• In the sub-atomic world the wavelength is significant and measurable: for electrons and protons it is of the order of the spacing of atomic planes in crystals, because their masses, and so their momenta, are so small.
• Exercise 11.10: a 0.040 kg bullet at 1.0 km/s has λ = 1.66 × 10⁻³⁵ m; a 0.060 kg ball at 1.0 m/s has λ = 1.1 × 10⁻³² m; a dust particle of 1.0 × 10⁻⁹ kg at 2.2 m/s has λ = 3.0 × 10⁻²⁵ m. All far too small to observe.
• The wavelength of a matter wave has physical meaning. Its phase velocity does not, but its group velocity does, and it equals the particle's velocity.

Teaching this idea? Put the simulation on your own page or LMS with this code.