The Photoelectric Effect and the Photon

Planck quantized the oscillators in a cavity wall but held that light itself was a continuous wave. Einstein removed the qualification: the quantization is a property of the radiation field. His test case was a laboratory curiosity Hertz had stumbled on in 1887 while confirming Maxwell's wave theory — light striking a metal surface knocks electrons loose. The wave theory that predicted the light also failed to predict what the ejected electrons do. Reconciling the two required treating the beam as a stream of particles.

The measurement

Illuminate a clean metal cathode and the electrons it emits — photoelectrons — can be collected at an anode, driving a current through an external circuit. A variable voltage $V$ between cathode and anode controls how many electrons arrive: a positive anode attracts them, a negative anode repels them.

Sweeping $V$ produces the current-voltage curve. At large positive $V$ every emitted electron is collected and the current saturates. As $V$ turns negative, only electrons with kinetic energy above $e|V|$ can climb the retarding potential and reach the anode. Below a certain negative value the current stops entirely.

Two features of these curves broke the wave theory. First, raising the intensity raises the saturation current — twice the light, twice the electrons — but leaves $V_0$ unchanged. Brighter light does not make the electrons faster.

Second, changing the frequency shifts $V_0$: bluer light gives a larger stopping potential, and below a threshold frequency $f_t$ no electrons are emitted at all, no matter how intense the light or how long it shines.

Classically neither should happen. A wave deposits energy at a rate set by intensity, so the electron's energy should grow with brightness, and any frequency of light, given enough time, should eventually shake an electron loose. There should also be a measurable time lag while a low-intensity wave slowly accumulates the escape energy in one atom. None of these classical expectations survived.

Einstein's photon

Einstein applied Planck's quantum to the light itself. A beam of frequency $f$ is not a smooth wave but a stream of discrete quanta, later called photons, each carrying energy

A photon is absorbed by a single electron all at once, delivering its whole energy $hf$. Freeing the electron from the metal costs a minimum energy $\varphi$, the work function, characteristic of the surface. Whatever remains appears as kinetic energy.

Energy conservation gives Einstein's photoelectric equation:

Every anomaly follows at once.

• Stopping potential independent of intensity. Intensity is the number of photons per second, not the energy each carries. More photons free more electrons — a larger current — but each electron still receives one photon's $hf$, so the maximum energy, and $V_0$, is fixed by frequency alone.
• Threshold frequency. A photon with $hf<\varphi$ cannot pay the escape cost, and the electron is not emitted regardless of how many such photons arrive. Setting $V_0=0$, $$\varphi =h{f}_t=\frac{hc}{{\lambda }_t}.$$
• No time lag. The energy is not accumulated gradually; a single absorption event delivers the whole $hf$ instantly, so emission can begin the moment the light is switched on, even at intensities where a classical wave would need minutes.

Millikan's confirmation

Einstein's equation is a straight line. Plotting $V_0$ against frequency $f$,

predicts a line of slope $h/e$ — the same for every metal — and intercept $-\varphi /e$ on the voltage axis, or $f_t=\varphi /h$ on the frequency axis. Millikan, who disbelieved the photon and set out to disprove the equation, instead confirmed it in 1914-1916 and measured $h$ from the slope in agreement with Planck's blackbody value.

The measurement was doubly important. It confirmed the photon, and it gave an independent value of Planck's constant from a phenomenon that, on its face, has nothing to do with cavity radiation. Two unrelated experiments returning the same $h$ is strong evidence the constant is real.

Worked examples

The convenient combination for photon energies is $hc=1240\text{eV}\cdot \text{nm}$, so a photon of wavelength $\lambda$ (in nm) carries $1240/\lambda$ eV.

Threshold and stopping potential for potassium. Potassium has threshold wavelength ${\lambda }_t=558\text{nm}$. The work function is the threshold photon energy,

Illuminating with $400\text{nm}$ light, each photon carries $1240/400=3.10\text{eV}$, so the stopping potential is

The classical time lag that never appears. Take $400\text{nm}$ light at a low intensity $I=10^{-2}{\text{W/m}}^2$, and let a target atom present a disk of radius $r\approx 10^{-10}\text{m}$. Classically the energy accumulates at

Reaching $\varphi =2.22\text{eV}=3.55\times 10^{-19}\text{J}$ would take

Classical physics predicts nearly a twenty-minute wait before the first electron appears. Experiment shows no delay. In the photon picture the same intensity is $2\times 10^{16}$ photons per second per square meter — about one photon per thousand surface atoms per second — and each carries enough energy to eject an electron immediately. Emission is prompt but sparse, not delayed and uniform.

The wave-particle tension

The photoelectric effect established that light exchanges energy with matter in discrete quanta $hf$, each localized enough to be absorbed by a single electron. Yet the same light diffracts and interferes, phenomena only a wave explains. Both descriptions are needed: a particle theory for the energy exchange, a wave theory for propagation and interference.

The photon carries energy $hf$; the next lesson shows it also carries momentum $hf/c$, and that a photon can scatter off an electron like one billiard ball off another. This Compton scattering of X-rays removed the last doubt that light is granular. The full reconciliation — that particles are also waves — waits for de Broglie and the uncertainty principle.¹²