Dual behaviour of matter and uncertainty principle

de Broglie (1924): matter, like radiation, is dual; a particle of mass m and speed v has wavelength λ = h/mv = h/p.

Electron beams diffract, a wave property; the electron microscope uses this and magnifies about 15 million times.

Every moving object has a wavelength, but for ordinary masses it is far too short to detect: a 0.1 kg ball at 10 m s⁻¹ has λ = 6.626 × 10⁻³⁴ m, while an electron with kinetic energy 3.0 × 10⁻²⁵ J (v ≈ 812 m s⁻¹) has λ ≈ 897 nm.

Heisenberg (1927): exact position and exact momentum of an electron cannot both be known at once; Δx·Δp ≥ h/4π, equivalently Δx·Δv ≥ h/4πm.

To locate an electron you must hit it with very short-wavelength light, whose high-momentum photons change the electron's velocity; the act of measuring disturbs it.

So an electron has no definite path or trajectory, and exact statements about it are replaced by probabilities.

The principle matters only for tiny masses: locating an electron within 0.1 Å leaves Δv ≈ 5.79 × 10⁶ m s⁻¹, but for a 40 g ball at 45 m s⁻¹ measured to 2%, Δx ≈ 1.46 × 10⁻³³ m, a meaningless limit.

Bohr's model fails because it ignores the wave nature of the electron and assumes a fixed orbit, a path that needs exact position and velocity together.