The photon

The photon is the quantum of light: the smallest indivisible packet of electromagnetic energy that can be exchanged at a given frequency. When light interacts with matter — being emitted, absorbed or scattered — it does not release energy continuously, but one photon at a time. It is this that makes light “corpuscular”.

Each photon carries an energy proportional to the frequency $f$ of the associated wave, and a momentum proportional to the inverse of the wavelength $\lambda$. The proportionality constant is Planck’s constant $h = 6{,}63\cdot 10^{-34}\;\text{J}\cdot\text{s}$, the quantity that fixes the “granularity” of the quantum world.

Key formula

$\ev{E = h\,f = \hbar\,\omega} \qquad \ev{|\vv{p}| = \frac{h}{\lambda} = \frac{h\,f}{c} = \frac{E}{c}}$ with $\hbar = h/(2\pi)$ (reduced Planck constant, “h-bar”), $\omega = 2\pi f$ the angular frequency, $f$ the frequency and $\lambda$ the wavelength.

Energy therefore grows with frequency: a photon of blue light is more energetic than one of red light, and an X-ray photon enormously more so. It is this scale that explains why high-frequency radiation (ultraviolet, X-rays, gamma rays) is ionising and potentially harmful, while low-frequency radiation (radio, infrared) is not: what matters is the energy of the single packet, not the overall intensity.

The relation $|\vv{p}| = E/c$ deserves attention: the photon has momentum despite having no mass. It is the ultra-relativistic limiting case of the energy-momentum relation, in which the energy is entirely “of motion” and none is “of rest”. A photon striking a surface transfers momentum to it, and this is the microscopic mechanism of radiation pressure.

Further insight — Consistency check with the wave picture

For an electromagnetic wave, Maxwell’s classical theory predicts a radiation pressure $P_\text{rad} = |I|/c$ (pressure = intensity divided by $c$). Dimensionally the intensity $|I|$ is energy per unit time and area, while $P_\text{rad}$ is momentum per unit time and area: their ratio is a speed, and it is precisely $c$. For a single photon this translates into the relation $\frac{E}{c} = p$ which is exactly what the corpuscular formula $p = h/\lambda = hf/c$ predicts. The wave description (Maxwell) and the corpuscular description (photons) lead to the same result: an early, elegant confirmation of the internal consistency of duality.

Links

Topics: Quantum physics Concepts: Photon · Wave-particle duality · Momentum · Radiation pressure

Related exercises: Connecting Malus’s law and a single photon · Problem — Collapse of the wave function at the polariser · Problem — Radiation pressure and comet tails