Matter waves and the de Broglie wavelength

Light had already forced the issue. A photon's energy and momentum obey E = pc, and E = hf with f = c/λ, so p = hf/c = h/λ — a momentum expressed entirely by a wavelength. In 1924 de Broglie asked which half of that statement is a fact about light. His hypothesis: neither. The relation belongs to momentum itself, so any particle of momentum p is accompanied by a wave of wavelength λ = h/p, whose frequency is f = E/h. Notice what the hypothesis does not claim. It does not say what is waving, and it does not say the wave travels at the particle's speed — the phase speed fλ = E/p is not v, and it is the group speed of a packet of such waves that keeps pace with the particle. What it does claim is testable: fix p and the diffraction angles are fixed with it.

Work in momentum, not speed. For mass m and kinetic energy K well below mc², p = √(2mK), so λ = h/√(2mK): the wavelength falls as 1/√K and as 1/√m. An electron accelerated from rest through a p.d. V gains K = eV, which gives the shortcut λ = 1.226 nm/√V with V in volts — 0.137 nm at 80.0 V, 0.100 nm at 150 V, 5.48 pm at 50.0 kV. A neutron is 1839 times heavier, so at the same kinetic energy its wavelength is √1839 = 42.9 times shorter: at 25 meV an electron has 7.76 nm against the neutron's 0.181 nm. A 0.160 kg ball at 30 m s⁻¹ carries p = 4.80 kg m s⁻¹, so λ = 6.626 × 10⁻³⁴ ÷ 4.80 = 1.4 × 10⁻³⁴ m. The scale is set by h, and h is tiny: matter wavelengths reach atomic size only for light particles moving slowly.

A wave announces itself only through interference, and that needs structure comparable to λ. At 0.1 nm no manufactured slit will serve, but a crystal is a ready-made grating with atoms about 0.2 nm apart. Davisson and Germer fired 54 eV electrons at a nickel crystal in 1927 and found a sharp peak in the scattered intensity at 50° from the incident beam, where classical scattering predicts a smooth falloff. The surface rows are d = 0.215 nm apart and constructive interference needs d sinφ = λ, giving λ = 0.215 × sin 50° = 0.165 nm. De Broglie's formula predicts 1.226/√54 = 0.167 nm. Two independent numbers agreeing to about 1% is what turned a hypothesis into physics. G. P. Thomson got there the other way, firing electrons through thin metal foils and photographing concentric rings identical in form to X-ray powder patterns. Neutrons, helium atoms and whole C₆₀ molecules (λ = 2.8 pm at 200 m s⁻¹) have been diffracted since.

A wave confined to a region can only take the wavelengths that fit, and λ = h/p turns that geometry into a statement about momentum. Trap an electron between walls a distance L apart: the standing waves need λ = 2L/n for n = 1, 2, 3 …, so p = nh/2L and K = n²h²/(8mL²). For L = 0.50 nm the lowest level is (6.626 × 10⁻³⁴)² ÷ (8 × 9.109 × 10⁻³¹ × (0.50 × 10⁻⁹)²) = 2.41 × 10⁻¹⁹ J = 1.50 eV — an electronvolt-sized gap conjured from nothing but a length and a fitting condition. Wrap the same argument round a circle and a Bohr orbit must hold a whole number of wavelengths, 2πr = nλ, which is exactly the angular-momentum rule mvr = nh/2π that Bohr had to assume. At the first Bohr radius 2πr = 0.3325 nm, and h/mv for that electron is also 0.3325 nm. Quantisation stops looking like an extra postulate and starts looking like a boundary condition.

An opening of width D spreads a beam by roughly θ ≈ λ/D radians, and everything classical follows from that ratio going to zero. The 0.160 kg ball has λ = 1.4 × 10⁻³⁴ m, so a 1 m doorway spreads it by 10⁻³⁴ rad, which is not merely small but unmeasurable — more than twenty orders of magnitude below any angle ever resolved. Even a 1 μg grain of dust drifting at 1 mm s⁻¹ has λ = 6.6 × 10⁻²² m. The classical limit is therefore not a different physics: it is λ/D → 0, in the same way geometrical optics is the λ → 0 limit of wave optics and rays replace wavefronts. Read the other way it is a design rule. A 50 kV electron has λ = 5.36 pm, about 10⁵ times shorter than green light, which is why an electron microscope resolves atomic columns — and why, inside its column, those same electrons can be traced as rays.

λ = h/p is exact; p = mv and p = √(2mK) are not. Once K is a noticeable fraction of the rest energy mc² — 511 keV for an electron — use E² = (pc)² + (mc²)² with E = K + mc², which gives pc = √(K² + 2Kmc²) and so λ = hc/√(K² + 2Kmc²), with hc = 1239.8 eV nm a convenient constant. The correction is invisible on a bench and decisive in an instrument: at 1.0 kV the non-relativistic wavelength is 0.05% too large, at 10 kV 0.5%, at 50 kV 2.4% (5.485 pm against the true 5.355 pm) and at 200 kV 9.3%. A 50 kV electron is already travelling at 0.413c. Nothing about the wave hypothesis changes here — only the arithmetic that converts an accelerating voltage into a momentum.