Where physics stops being tidy: light behaves like a stream of particles, and matter — electrons included — behaves like a wave. Welcome to the start of quantum physics.
Electrons in a metal are held in place by the metal's positive ions, needing a minimum energy — the work function φ₀ — to escape entirely. That energy can be supplied in several ways:
Shine light on certain metal surfaces and electrons (photoelectrons) are emitted immediately. Careful experiments (Hallwachs, Lenard, and others) revealed a set of results that classical wave theory simply couldn't explain:
Classical wave theory treats light's energy as spread continuously across the wavefront, so a brighter (more intense) wave should deliver more energy to each electron — predicting that kinetic energy should increase with intensity. It also predicts that, given enough time, even dim, low-frequency light should eventually eject an electron. Both predictions are flatly wrong: KEmax depends only on frequency, there's a hard threshold frequency below which nothing happens regardless of intensity or exposure time, and emission is instant rather than delayed. This mismatch was the crack that ultimately forced physics to accept light's particle nature.
Einstein resolved the puzzle by proposing that light itself arrives in discrete packets of energy — quanta — later named photons. Each photon delivers all its energy to a single electron in one shot, instantly, explaining the lack of delay.
A photon behaves like a genuine particle in every measurable sense — it carries definite energy and momentum, and interacts with electrons in discrete, all-or-nothing collisions.
If light — usually thought of as a wave — can behave like a particle, Louis de Broglie proposed the reverse should also hold: matter, usually thought of as particles, should also have an associated wave nature.
De Broglie's hypothesis was purely theoretical until Davisson and Germer fired a beam of electrons at a nickel crystal and observed a diffraction pattern — the unmistakable signature of wave behaviour, identical in kind to X-ray diffraction off the same crystal. The measured wavelength matched de Broglie's formula precisely, giving matter waves solid experimental footing and helping establish quantum mechanics as a whole.
The work function of a metal is 2.14 eV. Find its threshold wavelength.
Solution: λ₀ = hc/φ₀ = (6.63×10⁻³⁴ × 3×10⁸) / (2.14 × 1.6×10⁻¹⁹) = (1.989×10⁻²⁵) / (3.424×10⁻¹⁹) ≈ 5.81×10⁻⁷ m = 581 nm.
Find the de Broglie wavelength of an electron accelerated through a potential difference of 100 V.
Solution: Using λ ≈ 1.227/√V nm = 1.227/√100 = 1.227/10 ≈ 0.123 nm.
Notice this is comparable to atomic spacings in a crystal — exactly why electron diffraction off a crystal lattice (Davisson–Germer) was observable at all.