# Planck–Einstein relation

The Planck relation (referred to as Planck's energy–frequency relation, the Planck relation, Planck equation, and Planck formula, though the latter might also refer to Planck's law) is a fundamental equation in quantum mechanics which states that the energy of a photon, E, known as photon energy, is proportional to its frequency, ν:

$E=h\nu$ The constant of proportionality, h, is known as the Planck constant. Several equivalent forms of the relation exist, including in terms of angular frequency, ω:

$E=\hbar \omega$ where $\hbar =h/2\pi$ . The relation accounts for the quantized nature of light and plays a key role in understanding phenomena such as the photoelectric effect and black-body radiation (where the related Planck postulate can be used to derive Planck's law).

## Spectral forms

Light can be characterized using several spectral quantities, such as frequency ν, wavelength λ, wavenumber ${\tilde {\nu }}$ , and their angular equivalents (angular frequency ω, angular wavelength y, and angular wavenumber k). These quantities are related through

$\nu ={\frac {c}{\lambda }}=c{\tilde {\nu }}={\frac {\omega }{2\pi }}={\frac {c}{2\pi y}}={\frac {ck}{2\pi }},$ so the Planck relation can take the following 'standard' forms

$E=h\nu ={\frac {hc}{\lambda }}=hc{\tilde {\nu }},$ as well as the following 'angular' forms,

$E=\hbar \omega ={\frac {\hbar c}{y}}=\hbar ck.$ The standard forms make use of the Planck constant h. The angular forms make use of the reduced Planck constant ħ = h/. Here c is the speed of light.

## de Broglie relation

The de Broglie relation, also known as the de Broglie's momentum–wavelength relation, generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation E = would also apply to them, and postulated that particles would have a wavelength equal to λ = h/p. Combining de Broglie's postulate with the Planck–Einstein relation leads to

$p=h{\tilde {\nu }}$ or
$p=\hbar k.$ The de Broglie's relation is also often encountered in vector form

$\mathbf {p} =\hbar \mathbf {k} ,$ where p is the momentum vector, and k is the angular wave vector.

## Bohr's frequency condition

Bohr's frequency condition states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference (ΔE) between the two energy levels involved in the transition:

$\Delta E=h\nu .$ This is a direct consequence of the Planck–Einstein relation.