# Planck–Einstein relation

(Redirected from Planck–Einstein equation)

The Planck–Einstein relation[1][2] is also referred to as the Einstein relation,[1][3][4] Planck's energy–frequency relation,[5] the Planck relation,[6] and the Planck equation.[7] Also the eponym 'Planck formula'[8] belongs on this list, but also often refers instead to Planck's law[9][10] 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) 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.

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 forms

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

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

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

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

as well as the following 'angular' forms,

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

The angular forms make use of the reduced Planck constant $\hbar = \frac{h}{2 \pi}$. Here $c$ is the speed of light.

## de Broglie relation

The de Broglie relation,[4][11][12] also known as the de Broglie's momentum–wavelength relation,[5] generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation $E = h \nu$ would also apply to them, and postulated that particles would have a wavelength equal to $\lambda = \frac{h}{p}$. 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 $\mathbf{p}$ is the momentum vector, and $\mathbf{k}$ 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 ($\Delta E$) between the two energy levels involved in the transition:[13]

$\Delta E = h \nu.$

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

## References

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