Dual behaviour and uncertainty

Light behaves both as a wave and as particles (photons). In 1924, de Broglie suggested that the opposite is also true: every moving particle has a wave nature, with wavelength

λ = h/mv = h/p, where p = mv is the momentum.

This was confirmed when electrons were seen to be diffracted by crystals, just like X-rays. It led to the electron microscope, which uses electron waves to see far smaller things than ordinary light can.

Why don't we notice the wave nature of everyday objects? Because their wavelength is tiny. A 0.1 kg ball moving at 10 m s⁻¹ has λ = 6.626 × 10⁻³⁴ ÷ (0.1 × 10) = 6.6 × 10⁻³⁴ m, far too small for any instrument. For an electron, with its very small mass, λ is about the size of an atom, so it matters a great deal.

Heisenberg's uncertainty principle (1927): it is impossible to know both the exact position and the exact momentum (or velocity) of a particle at the same time:

Δx × Δp ≥ h/4π, or Δx × mΔv ≥ h/4π.

The more exactly we fix one, the less exactly we can know the other. For an electron located within 0.1 Å, the uncertainty in its velocity is about 5.8 × 10⁶ m s⁻¹, enormous. For a heavy object, the uncertainty is far too small to matter.

So we cannot say that an electron follows a definite orbit, as Bohr's model claimed. We can only give the probability of finding it in a particular place. Bohr's model ignored both the wave nature of the electron and the uncertainty principle.