Photoelectric effect through to the electron's own wavelength

The order tells the story of 1905 and 1924 in four lines. Step 1 chops a continuous beam into photons; step 2 spends one of them on one electron; step 3 reads the leftovers as ordinary motion; step 4 hands the wave nature back to the electron. The step that must come second is the subtraction, and the reason is the thing Einstein saw: an electron cannot save up. A faint beam of 5.06 eV photons frees electrons immediately, while an arbitrarily bright beam of 2.00 eV photons frees none at all, ever, no matter how long you leave it running. That threshold behaviour is what no wave theory could produce, and it is why the work function is subtracted from a single photon's energy rather than from the beam's.

Two traps, both bookkeeping. The first is the electron volt: 3.10 eV has to become 4.967 × 10⁻¹⁹ J before it can be subtracted from anything, and mixing the two units is the single most common way to lose the question. The second is confusing the photon's 245 nm with the electron's 0.876 nm at the end. They are wavelengths of entirely different things, and the electron's is roughly 280 times shorter even though the electron carries barely a third of the photon's energy — because λ = h/mv depends on momentum, and a slow massive particle has far more momentum per joule than a photon does. That mismatch is not a curiosity; it is the reason an electron microscope resolves atoms and an optical one never will.