The de Broglie hypothesis

The relation begins on the massless branch. Put m = 0 into E² = (pc)² + (mc²)² and you get E = pc exactly; combine that with E = hf and fλ = c, and a photon's momentum is p = E/c = hf/c = h/λ. This is not a definition dressed up as physics — Compton's 1923 measurement of the wavelength shift off nearly free electrons had just tested it against a two-body ledger, and it held. Read the chain in the direction it was derived and λ = h/p converts a known photon wavelength into a momentum. De Broglie's proposal was that the equation is more fundamental than the derivation that produced it: that λ = h/p is a statement about any quantum object, and that for an electron you should run it backwards, feeding in a momentum you already know how to compute and reading out a wavelength nobody had thought to look for. Nothing in the photon argument forces this step. It is a hypothesis, and it needed Davisson and Germer to survive.

De Broglie did not argue by analogy; he argued about phase. Attach to a particle of rest mass m an internal oscillation at its rest-frame frequency f₀ = mc²/h — for an electron, 1.24 × 10²⁰ Hz. In a lab frame where the particle moves at v, that internal clock is time-dilated and ticks at f₀/γ. Separately, let a wave of frequency f = γf₀ = E/h run through space, since it is the energy that transforms like a frequency. These two rates disagree, and de Broglie demanded that they agree at one place: the phase of the travelling wave evaluated at the particle's position x = vt must equal the phase of the internal clock. Writing that out gives γf₀ − v/λ = f₀/γ, so v/λ = f₀(γ − 1/γ) = f₀γβ², and therefore λ = c/(f₀γβ) = h/(γmv) = h/p. The same construction hands you the phase velocity: vₚₕₐₛₑ = fλ = E/p = c²/v, which is greater than c. That is not a contradiction, but it is a warning — the thing that actually moves at the particle's speed is the group velocity, and separating the two is the business of the wave-packet topic later in this unit.

The momentum you can actually control in a laboratory is the one an accelerating voltage supplies. An electron released from rest through a potential difference V arrives with kinetic energy K = eV, and while eV stays far below mₑc² = 511 keV that means p = √(2mₑeV). So λ = h/√(2mₑeV), and folding the constants together gives the number worth memorising: λ = 1.226 nm ÷ √(V in volts). Three checks. At V = 150 V, √150 = 12.25, so λ = 0.100 nm — one atomic spacing, and the reason a crystal is the natural grating for slow electrons. At the 54 V of the Davisson–Germer peak, λ = 1.226/7.348 = 0.167 nm. At V = 10 kV, λ = 1.226/100 = 12.3 pm, some forty times smaller than a 0.5 nm lattice spacing. Notice the exponent: λ ∝ V^(−1/2), so quadrupling the voltage halves the wavelength while doubling it buys only a factor of 1.41. Wavelength is expensive in volts.

√(2mₑeV) is the non-relativistic momentum, and it starts to lie as soon as eV is a noticeable fraction of mₑc². The exact statement comes from the invariant: (pc)² = (K + mc²)² − (mc²)² = K² + 2Kmc², so p = √(2mK)⋅√(1 + K/2mc²), and λ = h/√(2mₑeV(1 + eV/2mₑc²)). For electrons the correction factor is √(1 + V ÷ 1.022 × 10⁶), with V in volts. Take the 100 kV of a transmission electron microscope. Non-relativistically λ = 1.2264/√(1.00 × 10⁵) nm = 3.878 pm. The correction factor is √(1 + 0.09785) = 1.0478, so the true wavelength is 3.878/1.0478 = 3.701 pm and the shortcut overstates it by 4.8%. Set that factor equal to 1.01 and you locate the one-percent line: eV = (1.01² − 1) × 1.022 MeV = 20.5 keV. Above roughly 20 kV the non-relativistic formula is no longer good enough for anything you would fit a lattice constant to.

The result that made the hypothesis famous is one line of algebra. Insist that the wave close on itself around a circular orbit — a whole number of wavelengths, 2πr = nλ — and substitute λ = h/mv: 2πr = nh/mv, so mvr = nh/2π = nħ. Bohr's angular-momentum postulate, assumed out of nowhere in 1913, drops out of a standing-wave condition. The numbers land exactly. The first Bohr orbit has r₁ = 52.92 pm and v₁ = 2.188 × 10⁶ m s⁻¹, so λ = h/mₑv₁ = 332.5 pm while 2πr₁ = 332.5 pm — one wavelength, closed, to four figures. Now the bad news. The three-dimensional theory keeps the spectrum and throws the picture away: hydrogen's ground state has l = 0, zero orbital angular momentum, so there is nothing circulating and no loop for a wave to wrap around. The real quantisation condition is that ψ be single-valued and normalisable in three dimensions, of which the ring is a one-dimensional caricature. Right answer, wrong reason.

λ = h/p attaches a single number to a particle, and it is worth being precise about how little that number says. It does not say what is waving — de Broglie's phase wave had no interpretation, and the probability-amplitude reading is Born's, two years later. It supplies no equation of motion: knowing a wavelength does not tell you how the wave propagates through a potential, which is the gap Schrödinger closed in 1926. And it is exact only for a definite momentum, which means a plane wave of infinite extent; a genuinely localised electron is a superposition of many momenta, so it carries a spread of wavelengths rather than one, and "the" de Broglie wavelength is only the centre of that spread. Finally, the relation is unforgiving about scale. A 0.145 kg baseball thrown at 40 m s⁻¹ carries momentum 5.8 kg m s⁻¹, so λ = 6.63 × 10⁻³⁴ ÷ 5.8 = 1.1 × 10⁻³⁴ m — about 10⁻¹⁹ of a nuclear diameter. There is no aperture in the universe that small, which is why classical mechanics survives the hypothesis intact.