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Mathematics for Physics

188 Photon Energy

The quantized energy of a particle of light, proportional to its frequency through the Planck relation.

Quantization of radiant energy. Electromagnetic radiation arrives in discrete photons whose energy is proportional to frequency. A photon is one quantum of electromagnetic radiation. Frequency is the number of field oscillations per second. This principle is used to compute the energy delivered in the photoelectric effect.

The Planck-Einstein relation is

\[ E = hf \]

where

  • \(E\) is the photon energy.
  • \(h\) is Planck’s constant.
  • \(f\) is the frequency.

Photon momentum. A photon has zero rest mass and still carries momentum inversely proportional to its wavelength. Wavelength is the distance between successive crests of the field. This principle is used to compute recoil in Compton scattering.

The photon momentum is

\[ p = \dfrac{h}{\lambda} \]

where

  • \(p\) is the photon momentum.
  • \(\lambda\) is the wavelength.

Massless energy-momentum coupling. Energy and momentum of a photon satisfy the massless relation. A massless particle is a field quantum with no rest frame. This principle is used to obtain momentum from energy without a rest mass.

The massless energy-momentum relation is

\[ E = pc \]

where

  • \(c\) is the speed of light.

The relativistic Doppler transformation. A boost changes the measured frequency and wavelength of the same light signal. A coordinate boost is a Lorentz transformation between inertial frames. This principle is used to compute the Doppler shift of incoming radiation.

188.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(E=hf\); \(p=\dfrac{h}{\lambda}\).
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(E=hf\); \(E=pc\); Doppler shift.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — Planck relation; photon momentum; Doppler transformation.
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — \(E=pc\); relativistic Doppler shift.
  6. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — massless energy-momentum coupling.