# Planck–Einstein relation

The Planck–Einstein relation[1][2][3] is also referred to as the Einstein relation,[1][4][5] Planck's energy–frequency relation,[6] the Planck relation,[7] and the Planck equation.[8] Also the eponym Planck formula[9] belongs on this list, but also often refers to Planck's law instead.[10][11] These various eponyms are far from standard; they are used only sporadically, neither regularly nor very widely. They refer to a formula integral to quantum mechanics, which states that the energy of a photon, E, known as photon energy, is proportional to its frequency, ν:

${\displaystyle E=h\nu }$

The constant of proportionality, h, is known as the Planck constant. Several equivalent forms of the relation exist.

The relation accounts for quantized nature of light, and plays a key role in understanding phenomena such as the photoelectric effect, and Planck's law of black body radiation. See also the Planck postulate.

## Spectral formsEdit

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/. Here c is the speed of light.

## de Broglie relationEdit

The de Broglie relation,[5][12][13] also known as the de Broglie's momentum–wavelength relation,[6] 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

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

Bohr's frequency condition[citation needed] 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.

## ReferencesEdit

1. ^ a b French & Taylor (1978), pp. 24, 55.
2. ^ Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.
3. ^ Kalckar 1985, p. 39.
4. ^ Messiah (1958/1961), p. 72.
5. ^ a b Weinberg (1995), p. 3.
6. ^ a b Schwinger (2001), p. 203.
7. ^ Landsberg (1978), p. 199.
8. ^ Landé (1951), p. 12.
9. ^ Griffiths, D.J. (1995), pp. 143, 216.
10. ^ Griffiths, D.J. (1995), pp. 217, 312.
11. ^ Weinberg (2013), pp. 24, 28, 31.
12. ^ Messiah (1958/1961), p. 14.
13. ^ Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.
14. ^ van der Waerden (1967), p. 5.