The relationship between radiation's momentum, energy, and wavelength is described by the principles of wave-particle duality and relativistic energy-momentum relations.
According to wave-particle duality, electromagnetic radiation exhibits both wave-like and particle-like properties. It can be described in terms of particles called photons, which are quanta of electromagnetic energy.
The momentum (p) of a photon is related to its wavelength (λ) by the following equation, known as the de Broglie relation:
p = h / λ
where h is the Planck constant (approximately 6.626 × 10^-34 joule-seconds). This equation states that the momentum of a photon is inversely proportional to its wavelength. Photons with shorter wavelengths have higher momenta, while photons with longer wavelengths have lower momenta.
The energy (E) of a photon is related to its frequency (ν) by the equation:
E = h * ν
where ν represents the frequency of the photon. The frequency is related to the wavelength through the speed of light (c) by the equation:
c = λ * ν
Combining these equations, we can express the energy of a photon in terms of its wavelength:
E = h * c / λ
This equation shows that the energy of a photon is inversely proportional to its wavelength. Photons with shorter wavelengths have higher energies, while photons with longer wavelengths have lower energies.
It is worth noting that these relationships hold for massless particles, such as photons. For particles with mass, the energy-momentum relation is described by Einstein's relativistic equation:
E^2 = (mc^2)^2 + (pc)^2
where m is the rest mass of the particle, c is the speed of light, E is the total energy, and p is the momentum. This equation includes a term related to the rest mass of the particle, which is not present for massless particles like photons.
In summary, the momentum and energy of a photon are inversely proportional to its wavelength. Shorter wavelengths correspond to higher momenta and energies, while longer wavelengths correspond to lower momenta and energies.