How long is a matter wave?

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.