Planck–Einstein relation

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

${\displaystyle \nu ={\frac {c}{\lambda }}=c{\tilde {\nu }}={\frac {\omega }{2\pi }}={\frac {c}{2\pi y}}={\frac {ck}{2\pi }},}$

so the Planck relation can take the following 'standard' forms

${\displaystyle E=h\nu ={\frac {hc}{\lambda }}=hc{\tilde {\nu }},}$

as well as the following 'angular' forms,

${\displaystyle E=\hbar \omega ={\frac {\hbar c}{y}}=\hbar ck.}$

The standard forms make use of the Planck constant h. The angular forms make use of the reduced Planck constant ħ = h/2π. Here c is the speed of light.

The de Broglie relation,[10][11][12] also known as the de Broglie's momentum–wavelength relation,[4] generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation E = hν 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

${\displaystyle p=h{\tilde {\nu }}}$ or

${\displaystyle p=\hbar k.}$

The de Broglie's relation is also often encountered in vector form

${\displaystyle \mathbf {p} =\hbar \mathbf {k} ,}$

where p is the momentum vector, and k is the angular wave vector.

Bohr's frequency condition[13] 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:[14]

${\displaystyle \Delta E=h\nu .}$

This is a direct consequence of the Planck–Einstein relation.

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