Einstein's Photoelectric Equation: Energy Quantum of Radiation

Einstein took Planck's quantum hypothesis one bold step further. In 1905, he proposed that light energy itself is not absorbed continuously the way a wave picture demands, but arrives in discrete packets — quanta of radiation, each carrying energy $h\nu$. In the photoelectric effect, a single electron absorbs a single quantum. If that quantum's energy exceeds the work function $\phi_0$ — the minimum energy the electron needs to escape the metal surface — the electron is ejected with the leftover energy as kinetic energy:

More tightly bound electrons emerge with less than this maximum. This single equation — Einstein's photoelectric equation — explains every observation that defeated the wave picture, cleanly and without extra assumptions:

• $K_{\text{max}}$ depends linearly on frequency $\nu$ and is independent of intensity — exactly what experiment shows. In Einstein's picture, the photoelectric effect is the absorption of one photon by one electron; the intensity of the beam (the number of photons arriving per second) is simply irrelevant to how much energy any single absorption event delivers.
• Because $K_{\text{max}}$ can never be negative, Eq. (11.2) itself predicts that emission is only possible when $h\nu > \phi_0$, i.e. when $\nu$ exceeds a threshold frequency

A larger work function means a higher threshold frequency — below $\nu_0$, no amount of light intensity or exposure time can free an electron, because a single sub-threshold photon simply never carries enough energy to do the job on its own.

• Since intensity is proportional to the number of photons arriving per unit time, more intensity (for $\nu > \nu_0$) simply means more electrons absorbing a quantum each second — which is exactly why photocurrent scales with intensity even though each electron's energy does not.
• Emission is instantaneous because the elementary process — one photon handing its entire energy to one electron in a single absorption event — has no waiting time built in. Low intensity does not delay emission; it only means fewer electrons get the chance to absorb a photon in a given second.