Massless particles (photons)

Photons and rest mass

A photon is a quantum of electromagnetic radiation. It has zero rest mass:

$m=0$

But it still carries energy and momentum. This is not a contradiction because relativistic energy is not only rest energy. For a massless particle, the energy-momentum relation becomes

$E^2=p^2c^2$

so

$E=pc$

for positive energy.

Photon energy

Quantum theory relates photon energy to frequency:

$E=hf$

or equivalently

$E=hbaromega$

where $h$ is Planck's constant and $hbar=h/(2pi)$. Higher-frequency light has higher photon energy.

This explains why ultraviolet photons can cause chemical damage while radio photons usually cannot: the energy per photon is much larger for higher frequency.

Photon momentum

Combining $E=pc$ with $E=hf$ gives

p=rac{E}{c}=rac{hf}{c}

Using $c=flambda$, this becomes

p=rac{h}{lambda}

Thus shorter-wavelength photons have larger momentum.

Why photons travel at $c$

A massive particle has energy

$E=gamma mc^2$

and cannot reach $c$ because $gamma$ would become infinite. A massless particle cannot have a rest frame. If a photon could be at rest, it would have $E=mc^2=0$, contradicting its observed energy. In special relativity, massless particles in vacuum travel at exactly $c$.

Radiation pressure

Photons carry momentum, so light can exert pressure. If light of intensity $I$ is completely absorbed by a surface, the radiation pressure is

P*{rad}=rac{I}{c}

If perfectly reflected, the momentum change doubles, giving

P*{rad}=rac{2I}{c}

Radiation pressure is small in everyday life but important for solar sails, comet tails, laser cooling, and astrophysical processes.

Photon collisions

Photons participate in energy and momentum conservation. In Compton scattering, an X-ray photon scatters from an electron and changes wavelength. The shift is

Deltalambda=rac{h}{m_ec}(1-cos heta)

This confirms that photons carry momentum $h/lambda$.

Pair production and annihilation

A high-energy photon can help create particle-antiparticle pairs if energy and momentum conservation are satisfied, usually in the presence of another body such as a nucleus. Electron-positron annihilation can produce photons:

ightarrow gamma+gamma$$ The photons carry away energy and momentum. ## Wave and particle language Photons are not tiny classical pellets. They are quantum excitations of the electromagnetic field. Classical wave optics emerges when many photons occupy coherent states or when photon counting is not important. Quantum optics is needed when individual detection events, correlations, or photon statistics matter. ## The big idea Massless particles such as photons have zero rest mass but nonzero energy and momentum. They obey $E=pc$, while quantum theory gives $E=hf$ and $p=h/lambda$. Photon momentum explains radiation pressure, scattering, and the energy-momentum accounting of light.