The photon: energy and momentum

A massless particle, yet with energy

The photon is the elementary grain of light. It is a strange particle: it has no mass, it always travels at the speed of light c, and yet it carries a well-defined energy and momentum.

Its energy

A photon's energy depends only on the frequency f of the light (or, equivalently, on its wavelength λ, since c = λf):

   E = h × f = h × c / λ

The shorter the wavelength, the more energetic the photon. This is why ultraviolet and X-rays, with short wavelengths, are dangerous, whereas radio waves, with long wavelengths, are harmless.

   wavelength    :  large <---------------------------> small
   photon energy :  low   <---------------------------> high

   radio    microwaves    IR   VISIBLE   UV    X-rays     gamma
   |----------|-----------|------|-------|------|---------|----->
   harmless                                          dangerous

The electronvolt: the right unit

In joules, a photon's energy is a tiny number (~10^-19 J). We prefer the electronvolt (eV):

   1 eV = 1,6 × 10^-19 J

A visible-light photon has an energy of the order of 2 to 3 eV — a much more meaningful magnitude, on the scale of atoms.

Its momentum

More surprising still: although it is massless, the photon has a momentum p (a "momentum", usually associated with p = mv for a massive object):

   p = h / λ

This is not an abstraction: light really pushes what it strikes. This is radiation pressure, used by solar sail projects — craft propelled by the mere push of the Sun's photons.

   photons from the Sun
     -> -> -> ->  ||  <- solar sail pushed
     -> -> -> ->  ||     (each photon gives p = h/λ)

Counting photons

An ordinary light bulb emits a colossal number of photons per second (of the order of 10^20), which explains why we never perceive their "grainy" character: light seems continuous to us, just as a jet of water seems continuous although it is made of molecules. But with sensitive detectors, we can actually count photons one by one.

In summary

The photon is a massless particle with energy E = hf = hc/λ and momentum p = h/λ. Short-wavelength photons are the most energetic (UV, X, gamma). We measure their energy in electronvolts, and their momentum manifests concretely as radiation pressure.