A simple experiment with a stubborn result
Take a clean metal plate in a vacuum and shine light on it. Electrons are knocked out of the surface — you can collect them as a current. This is the photoelectric effect, and its details refused to fit the wave theory of light.
A wave delivers energy smoothly, so wave physics predicted three things: any colour should eventually work if you make it bright enough or wait long enough; a brighter light should give more energetic electrons; and there should be a time-lag while the electron soaks up energy.
Experiment said the opposite on every count. Below a certain threshold frequency, nothing happens at all, no matter how blinding the light. Above it, electrons appear instantly, even in a dim beam. And the maximum kinetic energy of the electrons depends on the frequency (colour) of the light — not on its brightness. Brightness only changes how many electrons come out.
Einstein's photon
In 1905 Einstein took Planck's lumps literally. He proposed that light itself is quantized: a beam is a hail of particles, photons, each carrying a fixed energy set only by the frequency. Planck had quantized the walls; Einstein quantized the light in flight.
A photon's energy: proportional to frequency, inversely proportional to wavelength.
A convenient shortcut for light: the product hc equals about 1240 eV·nm, so a photon of wavelength \lambda (in nanometres) has energy 1240/\lambda electron-volts. Violet light near 400 nm carries about 3.1 eV per photon; red near 700 nm only about 1.8 eV.
Now the puzzle dissolves. One photon gives all its energy to one electron. A high-frequency photon is a hard single kick; a low-frequency photon is too soft, and piling on more soft photons (a brighter dim-colour beam) never helps, because they act one at a time. That is exactly why there is a threshold.
The photoelectric equation
To escape the metal, an electron must first pay an energy toll — the work function \varphi, the minimum energy to break free of the surface. Whatever the photon has left over becomes the electron's kinetic energy.
Einstein's photoelectric equation: photon energy minus the work function.
The threshold frequency: below it, hf is less than the toll and no electron escapes.
Experimenters measure K_{\max} with a stopping voltage: apply a reverse voltage until even the fastest electrons are turned back, so eV_{\text{stop}} = K_{\max}. Plotting V_{\text{stop}} against frequency gives a straight line of slope h/e — which is how Robert Millikan measured Planck's constant from Einstein's equation.
See it and work it
Worked example. Violet light of wavelength 400 nm falls on a metal with work function \varphi = 2.30 eV. Find the maximum kinetic energy of the ejected electrons, the stopping voltage, and the longest wavelength that still works.
- Photon energy: E = 1240/\lambda = 1240/400 = 3.10 eV.
- Subtract the toll: K_{\max} = 3.10 - 2.30 = 0.80 eV.
- Stopping voltage: eV_{\text{stop}} = K_{\max}, so V_{\text{stop}} = 0.80 V.
- Threshold wavelength: \lambda_0 = 1240/\varphi = 1240/2.30 \approx 539 nm. Green or redder light (longer \lambda) ejects nothing, however bright.
Einstein won the 1921 Nobel Prize for this, not for relativity. It is the first hard proof that light is genuinely particle-like — which raises the obvious next question: if a wave can be a particle, can a particle be a wave?