How are photon energy, frequency, and wavelength related?

In physics, the relationship between photon energy, frequency, and wavelength is fundamental to understanding electromagnetic radiation and processes like luminescence. These properties are linked by the following principles:

• Energy and Frequency: The energy ($E$) of a single photon is directly proportional to its frequency ($\nu$ or $f$). This relationship is expressed by the equation $E=h\nu$ (or $E=hf$), where $h$ is Planck’s constant.12
• Frequency and Wavelength: Frequency and wavelength ($\lambda$) are inversely proportional, related by the speed of light ($c$) through the equation $\nu =c/\lambda$.13
• Combined Relationship: By substituting the frequency equation into the energy formula, we find that energy is inversely proportional to wavelength: $E=hc/\lambda$.2

Consequently, photons with shorter wavelengths possess higher frequencies and higher energies, while those with longer wavelengths have lower frequencies and lower energies. In the context of luminescence, these relationships dictate the energy efficiency of emission processes, as energy may be dissipated as heat when secondary photons are emitted at different wavelengths than the primary exciting photons.124