Photons

Classical electromagnetism pictures light as a continuous wave — an oscillating electric and magnetic field that can carry any amount of energy. That picture works beautifully for interference and diffraction. It fails, however, when light interacts with matter at the atomic scale. The resolution, first forced on physics by Planck's analysis of blackbody radiation and consolidated by Einstein's explanation of the photoelectric effect, is that electromagnetic energy is delivered in indivisible lumps called photons.

The energy of a photon

A monochromatic light beam of frequency $\nu$ is composed of photons each carrying energy

$E = h\nu,$

where $h \approx 6.626 \times 10^{-34}\,\text{J·s}$ is Planck's constant. Equivalently, using angular frequency $\omega = 2\pi\nu$ and the reduced Planck constant $\hbar = h / (2\pi) \approx 1.055 \times 10^{-34}\,\text{J·s}$ ,

$E = \hbar\omega.$

These two expressions are identical; the choice between them is purely notational convenience.

Momentum without rest mass

A photon is massless yet carries momentum. Special relativity relates a particle's energy $E$ , momentum $p$ , and rest mass $m$ through $E^2 = (pc)^2 + (mc^2)^2$ . For $m = 0$ this gives $E = pc$ , so a photon's momentum is

$p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda},$

where $\lambda = c/\nu$ is the wavelength. Written with the wave vector $k = 2\pi/\lambda = \omega/c$ , the momentum becomes $p = \hbar k$ . This relation, like $E = \hbar\omega$ , pairs a particle quantity ( $p$ ) with a wave quantity ( $k$ ), and it was confirmed directly by Compton scattering.

The photon as a quantum of the electromagnetic field

It is tempting to picture a photon as a tiny ball of light travelling through space, but that image is misleading. A photon is a quantum excitation of a mode of the electromagnetic field. Each mode of frequency $\nu$ can hold $n = 0, 1, 2, \ldots$ photons, and the energy of the field in that mode is

$E_n = \left(n + \frac{1}{2}\right)h\nu.$

The extra $\tfrac{1}{2}h\nu$ is the zero-point energy — the field fluctuates even in the vacuum state $n = 0$ . For the purposes of counting energy exchanges with atoms, however, each absorption or emission event transfers exactly one quantum $h\nu$ , which is why the simpler formula $E = h\nu$ suffices for most introductory calculations.

Why discreteness matters

The quantization of light has two immediate consequences that continuous-wave theory cannot reproduce:

Threshold behavior. Because energy arrives in lumps of size $h\nu$ , a single photon either has enough energy to ionize an atom or eject an electron, or it does not. Increasing the intensity (more photons per second) does not help if each individual photon is below the threshold — a result confirmed by the photoelectric effect.
Single-photon detection. Detectors such as photomultiplier tubes register discrete "clicks," one per photon. The number of clicks per second grows with intensity, but each click is a whole event, never a fraction of one. This granularity is direct evidence that energy arrives discontinuously.

Both observations are incompatible with classical wave theory and are exactly what $E = h\nu$ predicts. The photon concept therefore represents one of the foundational departures from classical physics that defines the quantum era.