Summary

Photoelectric effect: the ejection of electrons from a metal surface when light of a sufficiently high frequency falls on it; first noticed by Hertz (1887), investigated qualitatively by Hallwachs (1888) and quantitatively by Lenard (1902).

Laws of photoelectric emission: photocurrent $\propto$ intensity (fixed frequency above threshold); maximum kinetic energy $K_{max}$ of photoelectrons depends only on frequency, never on intensity; a threshold frequency $\nu_0$ exists, below which no emission occurs at any intensity; emission is essentially instantaneous.

Work function $\phi_0$: minimum energy needed to free the most loosely bound electron from a metal surface. Threshold frequency $\nu_0=\phi_0/h$; threshold wavelength $\lambda_0=hc/\phi_0$.

Failure of the wave theory: the classical continuous-energy wave picture predicts $K_{max}$ should rise with intensity, predicts no sharp threshold frequency, and predicts a measurable time lag at low intensity -- all three contradicted by experiment.

Einstein's photoelectric equation: $h\nu=\phi_0+K_{max}$, i.e. $K_{max}=h\nu-\phi_0$; equivalently, using the stopping potential, $eV_0=h\nu-\phi_0$, giving the straight-line graph $V_0$ vs $\nu$ of universal slope $h/e$. Rests on Einstein's postulate that light is absorbed in discrete quanta (photons) of energy $E=h\nu$, one photon transferring its entire energy to one electron.

Photon: a particle-like packet of radiation with energy $E=h\nu$, momentum $p=h/\lambda$, travelling at speed $c$, uncharged, conserving energy and momentum in its interactions -- established alongside the wave behaviour (interference/diffraction/polarization) of light as wave-particle duality.

de Broglie hypothesis (1924): matter, symmetrically with radiation, has an associated wavelength $\lambda=h/p=h/mv$ (the de Broglie wavelength/matter wave); observable only for very light particles such as electrons, since $h$ is minute compared with the momentum of any macroscopic object. …